Generalized derivations and commuting additive maps on multilinear polynomials in prime rings
Let $R$ be a prime ring with characteristic different from $2, U$ be its right Utumi quotient ring, $C$ be its extended centroid, $F$ and $G$ be additive maps on $R$ , $f(x_1, ..., x_n)$ be a multilinear polynomial over $C$, and $I$ be a nonzero right ideal of $R$ . We obtain information about the...
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2016
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| author | De, Filippis V. Dhara, B. Scudo, G. Де, Філіппіс В. Дхара, B. Сцудо, Г. |
| author_facet | De, Filippis V. Dhara, B. Scudo, G. Де, Філіппіс В. Дхара, B. Сцудо, Г. |
| author_sort | De, Filippis V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:29:16Z |
| description | Let $R$ be a prime ring with characteristic different from $2, U$ be its right Utumi quotient ring, $C$ be its extended centroid, $F$ and $G$ be additive maps on $R$ , $f(x_1, ..., x_n)$ be a multilinear polynomial over $C$, and $I$ be a nonzero right ideal of $R$ .
We obtain information about the structure of $R$ and describe the form of $F$ and $G$ in the following cases:
$$(1) [(F^2 + G)(f(r_1, ..., r_n)), f(r_1, ..., r_n)] = 0$$ for all $r_1, . . . , r_n \in R$, where $F$ and $G$ are generalized derivations of $R$ ;
$$(2) [(F^2 + G)(f(r_1, ..., r_n)), f(r_1, ..., r_n)] = 0$$for all $r_1, ..., r_n \in I$, where $F$ and $G$ are derivations of $R$. |
| first_indexed | 2026-03-24T02:13:30Z |
| format | Article |
| fulltext |
UDC 512.5
V. De Filippis (Univ. Messina, Italy),
B. Dhara (Belda College, India),
G. Scudo (Univ. Messina, Italy)
GENERALIZED DERIVATIONS AND COMMUTING ADDITIVE MAPS
ON MULTILINEAR POLYNOMIALS IN PRIME RINGS
УЗАГАЛЬНЕНI ПОХIДНI ТА КОМУТУЮЧI АДИТИВНI ВIДОБРАЖЕННЯ
НА МУЛЬТИЛIНIЙНИХ ПОЛIНОМАХ У ПРОСТИХ КIЛЬЦЯХ
Let R be a prime ring with characteristic different from 2, U be its right Utumi quotient ring, C be its extended centroid,
F and G be additive maps on R, f(x1, . . . , xn) be a multilinear polynomial over C, and I be a nonzero right ideal of R.
We obtain information about the structure of R and describe the form of F and G in the following cases:
(1) [(F 2+G)(f(r1, . . . , rn)), f(r1, . . . , rn)] = 0 for all r1, . . . , rn \in R, where F and G are generalized derivations
of R;
(2) [(F 2 +G)(f(r1, . . . , rn)), f(r1, . . . , rn)] = 0 for all r1, . . . , rn \in I , where F and G are derivations of R.
Нехай R — просте кiльце з характеристикою, що вiдмiнна вiд 2, U — його праве фактор-кiльце, C — його розширений
центроїд, F та G — адитивнi вiдображення на R, f(x1, . . . , xn) — мультилiнiйний полiном над C, а I — ненульовий
правий iдеал для R. Отримано iнформацiю про структуру кiльця R та описано форму F i G у таких випадках:
(1) [(F 2 + G)(f(r1, . . . , rn)), f(r1, . . . , rn)] = 0 для всiх r1, . . . , rn \in R, де F та G — узагальненi похiднi
вiд R;
(2) [(F 2 +G)(f(r1, . . . , rn)), f(r1, . . . , rn)] = 0 для всiх r1, . . . , rn \in I , де F та G — похiднi вiд R.
1. Introduction. Throughout this paper, R always denotes a prime ring with center Z(R) and
extended centroid C, U its right Utumi quotient ring. By a derivation on R, we mean an additive map
G : R - \rightarrow R such that G(xy) = G(x)y + xG(y) holds for all x, y \in R. A generalized derivation on
R is an additive map G : R - \rightarrow R such that G(xy) = G(x)y + xd(y) holds for all x, y \in R, where
d is a derivation of R. We denote [a, b] = ab - ba, the simple commutator of the elements a, b \in R
and [a, b]k =
\bigl[
[a, b]k - 1, b
\bigr]
, for k > 1, the kth commutator of a, b. Let T \subseteq R. An additive map F :
R - \rightarrow R is said to be commuting in T (resp. centralizing in T ) if [F (x), x] = 0 for all x \in T (resp.
[F (x), x] \in Z(R) for all x \in T ).
Several authors have studied derivations and generalized derivations which are centralizing and
commuting in some subsets of prime and semiprime rings (see [12, 17, 19, 21] for references). In this
view, a well-known result proved by Posner [24] states that a prime ring R must be commutative, if
it admits a non-zero centralizing derivation. In [16], Lee studied derivations with Engel conditions
on polynomials f(x1, . . . , xn) in non-zero one-sided ideals of R. More precisely, he proved that if\bigl[
d
\bigl(
f(r1, . . . , rn)
\bigr)
, f(r1, . . . , rn)
\bigr]
k
= 0 for all r1, . . . , rn \in L, a non-zero left ideal of R, and k \geq 1
a fixed integer, then there exists an idempotent element e in the socle of RC such that CL = RCe
and one of the following holds: (i) f(x1, . . . , xn) is central valued in eRCe unless C is finite or
0 < \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(R) \leq k+1; (ii) in case \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(R) = p > 0, then f(x1, . . . , xn)
ps is central valued in eRCe
for some s \geq 0, unless \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(R) = 2 and eRCe satisfies the identity s4.
Recently in [6], the first author of the present paper studied the case when the Engel condition is
satisfied by a generalized derivation on evaluations of multilinear polynomials. More precisely, he
proved that if G is a non-zero generalized derivation of R, f(x1, . . . , xn) a multilinear polynomial
c\bigcirc V. DE FILIPPIS, B. DHARA, G. SCUDO, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2 183
184 V. DE FILIPPIS, B. DHARA, G. SCUDO
over C and I a non-zero right ideal of R such that G is commuting in f(I), the set of all evaluations
of f(x1, . . . , xn) over I, then either G(x) = ax with (a - \gamma )I = 0 and a suitable \gamma \in C or there
exists an idempotent element e \in \mathrm{s}\mathrm{o}\mathrm{c}(RC) such that IC = eRC and one of the following holds:
(1) f(x1, . . . , xn) is central valued in eRCe;
(2) G(x) = cx+ xb, where (c - b+ \alpha )e = 0 for \alpha \in C and f(x1, . . . , xn)
2 is central valued in
eRCe;
(3) \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(R) = 2 and s4(x1, x2, x3, x4) is an identity for eRCe.
Simultaneously in [1], Ali and Shah showed that if the generalized derivation G is centralizing in a
one-sided ideal of the prime ring R, then R is commutative.
The main object of the present paper is to investigate the situation when the additive map F 2+G
is commuting in f(I), the set of all evaluations of f(x1, . . . , xn) over I, where f(x1, . . . , xn) is a
multilinear polynomial over C, I is a suitable subset of R and F,G two derivations or generalized
derivations of R.
In [14] (Theorem 2.1), Lee et al. proved that if F and G are derivations of a n!-torsion free
semiprime ring such that [(F 2+G)(x), xn] = 0 for all x \in R, then F and G are both commuting in R.
Recently in [8], the first author of the present paper and Rehman extended the above result of [14]
to generalized derivations. More precisely, in [8] (Theorem 3.1), it is proved that if R is a n!-torsion
free semiprime ring, F and G two generalized derivations of R associated with non-zero derivations
f and g respectively, such that
\bigl[
(F 2 + G)(x), xn
\bigr]
= 0 for all x \in R, then either R contains a
non-zero central ideal, or f = 0, g(R) \subseteq Z(R) and there exist a, b \in U such that F (x) = ax,
G(x) = bx+ g(x) for all x \in R, with a2 + b \in C.
Recently in [9], the second author and Sharma studied the case when F is a derivation of R,
f(x1, . . . , xn) is a multilinear polynomial over C and I is a right ideal of R.
They proved that if
\bigl[
F 2
\bigl(
f(r1, . . . , rn)
\bigr)
, f(r1, . . . , rn)
\bigr]
= 0 for all r1, . . . , rn \in I, then either\bigl[
f(x1, . . . , xn), xn+1
\bigr]
xn+2 is satisfied by I, or there exists b \in U such that F (x) = [b, x] for all
x \in R, with b2 = 0 and bI = (0).
Being inspired by the above cited results, we shall prove the following theorem.
Theorem 1.1. Let R be a prime ring of characteristic different from 2, U its right Utumi
quotient ring, C its extended centroid, F and G two generalized derivations of R and f(x1, . . . , xn)
a multilinear polynomial over C. If
\bigl[
(F 2+G)
\bigl(
f(r1, . . . , rn)
\bigr)
, f(r1, . . . , rn)
\bigr]
= 0 for all r1, . . . , rn \in
\in R, then either f(x1, . . . , xn) is central valued on R or one of the following holds:
(1) there exist c, p \in U such that F (x) = xc, G(x) = xp for all x \in R, with c2 + p \in C;
(2) there exist c, p \in U and \alpha \in C such that F (x) = cx, G(x) = px for all x \in R, with
c2 + p \in C;
(3) f(x1, . . . , xn)
2 is central valued on R and there exist c, p, q \in U such that F (x) = xc,
G(x) = px+ xq for all x \in R, with c2 + q - p \in C;
(4) f(x1, . . . , xn)
2 is central valued on R and there exist c, p, q \in U such that F (x) = cx,
G(x) = px+ xq for all x \in R, with c2 + p - q \in C.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2
GENERALIZED DERIVATIONS AND COMMUTING ADDITIVE MAPS ON MULTILINEAR . . . 185
Theorem 1.2. Let R be a prime ring of characteristic different from 2, U its right Utumi quotient
ring, C its extended centroid, F and G two derivations of R, f(x1, . . . , xn) a multilinear polynomial
over C and I a non-zero right ideal of R. If
\bigl[
(F 2 + G)
\bigl(
f(r1, . . . , rn)
\bigr)
, f(r1, . . . , rn)
\bigr]
= 0 for all
r1, . . . , rn \in I, then one of the following holds:
(1) there exists an idempotent element e \in \mathrm{s}\mathrm{o}\mathrm{c}(RC) such that IC = eRC,moreover f(x1, . . . , xn)
is central valued on eRCe;
(2) there exist c, p \in U and \alpha , \beta \in C such that F (x) = [c, x], G(x) = [p, x] for all x \in R, with
(c - \alpha )I = (p - \beta )I = (0) and (c - \alpha )2 = (p - \beta ).
To prove our theorems, we shall use frequently the theory of generalized polynomial identities
and differential identities (see [2, 4, 13, 20, 23]). In particular, we recall that if R is prime and I a
non-zero right ideal of R, then I, IR and IU satisfy the same generalized polynomial identities [4].
In [17], Lee extended the definition of a generalized derivation as follows: by a generalized
derivation we mean an additive mapping g : I - \rightarrow U such that g(xy) = g(x)y + xd(y) for all
x, y \in I, where I is a dense right ideal of R and d is a derivation from I into U.
Moreover, Lee also proved that every generalized derivation can be uniquely extended to a
generalized derivation of U and thus all generalized derivations of R will be implicitly assumed to
be defined on the whole U.
More details about generalized derivations can be found in [11, 17, 21].
2. The case: pair of generalized derivations on multilinear polynomials in prime rings. In
this section we will prove Theorem 1.1. We begin with the following lemma, which will be also used
in the next section for the proof of Theorem 1.2.
Lemma 2.1. Let R be a prime ring, F (x) = ax + xb and G(x) = px + xq, for a, b, p, q \in
\in U, be two inner generalized derivations of R. Let I be a right ideal of R such that
\bigl[
(F 2 +
+ G)
\bigl(
f(r1, . . . , rn)
\bigr)
, f(r1, . . . , rn)
\bigr]
= 0 for all r1, . . . , rn \in I. Then R satisfies a nontrivial gene-
ralized polynomial identity, unless one of the following holds:
(1) there exist \alpha , \beta \in C such that (a2 + p - \alpha )I = (0), (a - \beta )I = (0) and b2 + q + 2\beta b \in C;
(2) b, q \in C and there exists \alpha \in C such that (a2 + p+ 2ab - \alpha )I = (0).
Proof. Let B be a basis of U over C. Then any element of T = U \ast C C\{ x1, . . . , xn\} can
be written in the form g =
\sum
i
\alpha imi. In this decomposition the coefficients \alpha i are in C and the
elements mi are B-monomials, that is mi = q0y1 . . . yhqh, with qi \in B and yi \in \{ x1, . . . , xn\} .
In [4], it is shown that a generalized polynomial g =
\sum
i
\alpha imi is the zero element of T if and
only if all \alpha i are zeros. As a consequence, let a1, . . . , ak \in U be linearly independent over C and
a1g1(x1, . . . , xn) + . . .+ akgk(x1, . . . , xn) = 0 \in T, for some g1, . . . , gk \in T.
If, for any i, gi(x1, . . . , xn) =
\sum n
j=1
xjhj(x1, . . . , xn) and hj(x1, . . . , xn) \in T, then g1(x1, . . .
. . . , xn), . . . , gk(x1, . . . , xn) are the zero elements of T. The same conclusion holds if g1(x1, . . .
. . . , xn)a1 + . . . + gk(x1, . . . , xn)ak = 0 \in T, and gi(x1, . . . , xn) =
\sum n
j=1
hj(x1, . . . , xn)xj for
some hj(x1, . . . , xn) \in T.
In all that follows we assume that R does not satisfy any nontrivial generalized polynomial
identity with coefficients in U. Therefore by our hypothesis, for any 0 \not = y \in I,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2
186 V. DE FILIPPIS, B. DHARA, G. SCUDO
\Phi (x1, . . . , xn) =
\Bigl[
(a2 + p)f(yx1, . . . , yxn) + 2af(yx1, . . . , yxn)b+
+f(yx1, . . . , yxn)(b
2 + q), f(yx1, . . . , yxn)
\Bigr]
is a trivial generalized polynomial identity for R. We rewrite it as
\Phi (x1, . . . , xn) = (a2 + p)f(yx1, . . . , yxn)
2 + 2af(yx1, . . . , yxn)bf(yx1, . . . , yxn)+
+f(yx1, . . . , yxn)(b
2 + q)f(yx1, . . . , yxn) - f(yx1, . . . , yxn)(a
2 + p)f(yx1, . . . , yxn) -
- 2f(yx1, . . . , yxn)af(yx1, . . . , yxn)b - f(yx1, . . . , yxn)
2(b2 + q). (2.1)
If b2 + q \in C, then by applying previous argument to (2.1), we have b \in C. Analogously if b \in C,
it follows b2 + q \in C. Hence b \in C if and only if b2 + q \in C. On the other hand, by applying
the same argument to (2.1), \{ (a2 + p)y, y\} is linearly C-dependent if and only if \{ ay, y\} is linearly
C-dependent. Now we divide the proof into three cases:
Case 1. Suppose that b2 + q, b \in C. Then by (2.1), it follows that
\bigl\{
(a2 + p + 2ab)y, y
\bigr\}
is
linearly C-dependent. Thus there exists \alpha \in C such that (a2 + p + 2ab - \alpha )I = (0), which is our
conclusion (2).
Case 2. Suppose that for any y \in I,
\bigl\{
(a2 + p)y, y
\bigr\}
as well as \{ ay, y\} are two linearly C-
dependent sets. In this case standard argument shows that there exist \alpha , \lambda \in C such that (a2 + p -
- \alpha )I = (0) and (a - \lambda )I = (0). Then (2.1) reduces to\bigl[
f(yx1, . . . , yxn)(2\lambda b+ b2 + q), f(yx1, . . . , yxn)
\bigr]
which is a trivial generalized polynomial identity for R, implying 2\lambda b+b2+q \in C. Thus conclusion (1)
is obtained.
Case 3. We denote u = a2 + p and v = b2 + q. Finally, suppose that b /\in C, b2 + q /\in C and
there exists y0 \in I such that \{ uy0, y0\} is linearly C-independent as well as \{ ay0, y0\} is linearly
C-independent. Since R does not satisfy any nontrivial generalized polynomial identity, by (2.1) we
have both the cases:
\{ b, v, 1\} is linearly C dependent, so that there exist \beta 1, \beta 2 \in C such that b = \beta 1v+\beta 2. Moreover
\beta 1 \not = 0, since b /\in C;
\{ uy0, ay0, y0\} is linearly C-dependent, so that there exist \alpha 1, \alpha 2 \in C such that uy0 = \alpha 1ay0 +
+ \alpha 2y0. Moreover \alpha 1 \not = 0, since uy0 \not = \alpha 2y0.
Hence by (2.1), R satisfies
(\alpha 1a+ \alpha 2)f(y0x1, . . . , y0xn)
2 + 2af(y0x1, . . . , y0xn)(\beta 1v + \beta 2)f(y0x1, . . . , y0xn)+
+f(y0x1, . . . , y0xn)vf(y0x1, . . . , y0xn) -
- f(y0x1, . . . , y0xn)(\alpha 1a+ \alpha 2)f(y0x1, . . . , y0xn) -
- 2f(y0x1, . . . , y0xn)af(y0x1, . . . , y0xn)(\beta 1v + \beta 2) - f(y0x1, . . . , y0xn)
2v
which implies that \{ \beta 1v + \beta 2, v, 1\} is linearly C-dependent. Since we assume v /\in C, it follows
\beta 1v + \beta 2 = 0 and so \beta 1 = 0, a contradiction.
Lemma 2.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2
GENERALIZED DERIVATIONS AND COMMUTING ADDITIVE MAPS ON MULTILINEAR . . . 187
An easy consequence of the previous result is the following lemma.
Lemma 2.2. Let R be a prime ring and F (x) = ax+ xb, G(x) = px+ xq, for a, b, p, q \in U be
two inner generalized derivations of R such that\bigl[
(F 2 +G)
\bigl(
f(r1, . . . , rn)
\bigr)
, f(r1, . . . , rn)
\bigr]
= 0
for all r1, . . . , rn \in R. Then R satisfies a nontrivial generalized polynomial identity, unless one of
the following holds:
(1) a2 + p \in C, a \in C and b2 + q + 2ab \in C;
(2) b, q \in C and a2 + p+ 2ab \in C.
Fact 2.1 (Theorem 1 in [6]). Let R be a prime ring, a, b \in R and f(x1, . . . , xn) a noncen-
tral multilinear polynomial over C. If
\bigl[
af(r1, . . . , rn) - f(r1, . . . , rn)b, f(r1, . . . , rn)
\bigr]
= 0 for all
r1, . . . , rn \in R, then one of the following conclusions holds:
(1) a, b \in Z(R);
(2) f(x1, . . . , xn)
2 is central valued on R and a+ b \in C;
(3) \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(R) = 2 and R satisfies the standard identity s4.
Lemma 2.3. Let R be a prime ring with \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r} (R) \not = 2 and f(x1, . . . , xn) a noncentral mul-
tilinear polynomial over C. Assume that F (x) = xb and G(x) = px + xq, for a, b, p, q \in U. If\bigl[
(F 2 + G)
\bigl(
f(r1, . . . , rn)
\bigr)
, f(r1, . . . , rn)
\bigr]
= 0 for all r1, . . . , rn \in R, then one of the following
holds:
(1) p \in C and b2 + q \in C;
(2) f(x1, . . . , xn)
2 is central valued on R and b2 + q - p \in C.
Proof. In this case we have that R satisfies the generalized identity\bigl[
pf(x1, . . . , xn) + f(x1, . . . , xn)(b
2 + q), f(x1, . . . , xn)
\bigr]
.
Hence the required result follows from Fact 2.1.
Lemma 2.4. Let R be a prime ring with \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r} (R) \not = 2 and f(x1, . . . , xn) a noncentral mul-
tilinear polynomial over C. Assume that F (x) = ax and G(x) = px + xq, for a, b, p, q \in U. If\bigl[
(F 2 + G)
\bigl(
f(r1, . . . , rn)
\bigr)
, f(r1, . . . , rn)
\bigr]
= 0 for all r1, . . . , rn \in R, then one of the following
holds:
(1) q \in C and a2 + p \in C;
(2) f(x1, . . . , xn)
2 is central valued on R and a2 + p - q \in C.
Proof. In this case R satisfies the generalized identity\bigl[
(a2 + p)f(x1, . . . , xn) + f(x1, . . . , xn)q, f(x1, . . . , xn)
\bigr]
and as above we get the required conclusion by applying again Fact 2.1.
Lemma 2.5 (Lemma 1 in [7]). Let C be an infinite field and m \geq 2. If A1, . . . , Ak are not
scalar matrices in Mm(C), then there exists some invertible matrix P \in Mm(C) such that any
matrix PA1P
- 1, . . . , PAkP
- 1 has all non-zero entries.
Proposition 2.1. Let R = Mm(C) be the ring of all (m \times m)-matrices over the infinite field
C and f(x1, . . . , xn) a noncentral multilinear polynomial over C. Assume that F (x) = ax + xb
and G(x) = px + xq, for a, b, p, q \in R. If
\bigl[
(F 2 + G)
\bigl(
f(r1, . . . , rn)
\bigr)
, f(r1, . . . , rn)
\bigr]
= 0 for all
r1, . . . , rn \in R, then one of the following holds:
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2
188 V. DE FILIPPIS, B. DHARA, G. SCUDO
(1) a, p \in C \cdot Im and (a+ b)2 + q \in C \cdot Im;
(2) b, q \in C \cdot Im and (a+ b)2 + p \in C \cdot Im;
(3) f(x1, . . . , xn)
2 is central valued on R, a \in C \cdot Im and (a+ b)2 + q - p \in C \cdot Im;
(4) f(x1, . . . , xn)
2 is central valued on R, b \in C \cdot Im and (a+ b)2 + p - q \in C \cdot Im.
Proof. By our assumption R satisfies the generalized identity\bigl[
(a2 + p)f(x1, . . . , xn) + 2af(x1, . . . , xn)b+ f(x1, . . . , xn)(b
2 + q), f(x1, . . . , xn)
\bigr]
. (2.2)
If either a \in Z(R) or b \in Z(R), then the conclusions follow by Lemmas 2.3 and 2.4 respectively.
Therefore we assume that a /\in Z(R) and b /\in Z(R). Now we shall show that this case leads a
contradiction.
Since a /\in Z(R) and b /\in Z(R), by Lemma 2.5 there exists an C-automorphism \varphi of Mm(C)
such that a\prime = \varphi (a), b\prime = \varphi (b) have all non-zero entries. Clearly a\prime , b\prime , p\prime = \varphi (p) and q\prime = \varphi (q)
must satisfy the condition (2.2). Without loss of generality we may replace a, b, p, q with a\prime , b\prime , p\prime , q\prime
respectively.
Here ekl denotes the usual matrix unit with 1 in (k, l)-entry and zero elsewhere. Since f(x1, . . . , xn)
is not central, by [20] (see also [22]), there exist u1, . . . , un \in Mm(C) and \gamma \in C - \{ 0\} such that
f(u1, . . . , un) = \gamma ekl, with k \not = l. Moreover, since the set \{ f(r1, . . . , rn) : r1, . . . , rn \in Mm(C)\}
is invariant under the action of all C-automorphisms of Mm(C), then for any i \not = j there exist
r1, . . . , rn \in Mm(C) such that f(r1, . . . , rn) = eij . Hence by (2.2) we have\bigl[
(a2 + p)eij + 2aeijb+ eij(b
2 + q), eij
\bigr]
= 0
and then right multiplying by eij , it follows 2eijaeijbeij = 0, which is a contradiction, since a and b
have all non-zero entries.
Proposition 2.1 is proved.
Proposition 2.2. Let R = Mm(C) be the ring of all matrices over the field C with \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r} (R) \not = 2
and f(x1, . . . , xn) a noncentral multilinear polynomial over C. Assume that F (x) = ax + xb
and G(x) = px + xq for a, b, p, q \in R. If
\bigl[
(F 2 + G)
\bigl(
f(r1, . . . , rn)
\bigr)
, f(r1, . . . , rn)
\bigr]
= 0, for all
r1, . . . , rn \in R, then one of the following holds:
(1) a, p \in C \cdot Im and (a+ b)2 + q \in C \cdot Im;
(2) b, q \in C \cdot Im and (a+ b)2 + p \in C \cdot Im;
(3) f(x1, . . . , xn)
2 is central valued on R, a \in C \cdot Im and (a+ b)2 + q - p \in C \cdot Im;
(4) f(x1, . . . , xn)
2 is central valued on R, b \in C \cdot Im and (a+ b)2 + p - q \in C \cdot Im.
Proof. If one assumes that C is infinite, then the conclusions follow by Proposition 2.1.
Now let C be finite and K be an infinite field which is an extension of the field C. Let R =
= Mm(K) \sim = R\otimes C K. Notice that the multilinear polynomial f(x1, . . . , xn) is central-valued on R
if and only if it is central-valued on R. Consider the generalized polynomial
P (x1, . . . , xn) =
=
\bigl[
(a2 + p)f(x1, . . . , xn) + 2af(x1, . . . , xn)b+ f(x1, . . . , xn)(b
2 + q), f(x1, . . . , xn)
\bigr]
which is a generalized polynomial identity for R.
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GENERALIZED DERIVATIONS AND COMMUTING ADDITIVE MAPS ON MULTILINEAR . . . 189
Moreover, it is a multihomogeneous of multidegree (2, . . . , 2) in the indeterminates x1, . . . , xn.
Hence the complete linearization of P (x1, . . . , xn) is a multilinear generalized polynomial
\Theta (x1, . . . , xn, y1, . . . , yn) in 2n indeterminates, moreover
\Theta (x1, . . . , xn, x1, . . . , xn) = 2nP (x1, . . . , xn).
Clearly the multilinear polynomial \Theta (x1, . . . , xn, y1, . . . , yn) is a generalized polynomial identity for
R and R too. Since \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(C) \not = 2 we obtain P (r1, . . . , rn) = 0 for all r1, . . . , rn \in R and then
conclusion follows from Proposition 2.1.
Proposition 2.2 is proved.
Fact 2.2 (Reduced version of Theorem 3.1 in [8]). Let R be a prime ring, F and G two genera-
lized derivations of R such that [(F 2 + G)(x), x] = 0 for all x \in R. Then one of the following
holds:
(1) R is commutative;
(2) there exist a, b \in U such that F (x) = ax and G(x) = bx for all x \in R, with a2 + b \in C;
(3) there exist a, b \in U such that F (x) = xa and G(x) = xb for all x \in R, with a2 + b \in C.
Proposition 2.3. Let R be a prime ring of characteristic different from 2 and f(x1, . . . , xn)
a noncentral multilinear polynomial over C. Assume that F (x) = ax + xb and G(x) = px + xq
for a, b, p, q \in U. If [(F 2 +G)(f(r1, . . . , rn)), f(r1, . . . , rn)] = 0 for all r1, . . . , rn \in R, then one of
the following holds:
(1) a, p \in C and (a+ b)2 + q \in C;
(2) b, q \in C and (a+ b)2 + p \in C;
(3) f(x1, . . . , xn)
2 is central valued on R, a \in C and (a+ b)2 + q - p \in C;
(4) f(x1, . . . , xn)
2 is central valued on R, b \in C and (a+ b)2 + p - q \in C.
Proof. By Lemma 2.2, we may assume that R satisfies the nontrivial generalized polynomial
identity
P (x1, . . . , xn) =
=
\bigl[
(a2 + p)f(x1, . . . , xn) + 2af(x1, . . . , xn)b+ f(x1, . . . , xn)(b
2 + q), f(x1, . . . , xn)
\bigr]
.
By a theorem due to Beidar (Theorem 2 in [3]) this generalized polynomial identity is also satisfied
by U. In case C is infinite, we have P (r1, . . . , rn) = 0 for all r1, . . . , rn \in U
\bigotimes
C C, where C is the
algebraic closure of C. Since both U and U
\bigotimes
C C are centrally closed [10] ( Theorems 2.5 and 3.5),
we may replace R by U or U
\bigotimes
C C according as C is finite or infinite. Thus we may assume that R
is centrally closed over C which is either finite or algebraically closed. By Martindale’s theorem [23],
R is a primitive ring having a non-zero socle H with C as the associated division ring and eHe is a
simple central algebra finite dimensional over C, for any minimal idempotent element e \in H.
In light of Jacobson’s theorem [12, p. 75], R is isomorphic to a dense ring of linear transformations
on some vector space V over C.
Assume first that V is finite-dimensional over C. Then the density of R on V implies that
R \sim = Mk(C), the ring of all (k \times k)-matrices over C. Since R is not commutative, we may assume
k \geq 2. In this case the conclusion follows by Proposition 2.2.
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190 V. DE FILIPPIS, B. DHARA, G. SCUDO
Assume next that V is infinite-dimensional over C. As in Lemma 2 in [25], the set f(R) is dense
on R and so from P (r1, . . . , rn) = 0 for all r1, . . . , rn \in R, we have that R satisfies the generalized
identity \bigl[
(a2 + p)x+ 2axb+ x(b2 + q), x
\bigr]
= 0
that is \bigl[
(a2 + p)r + 2arb+ r(b2 + q), r
\bigr]
= 0
for all r \in R. In this case, by Fact 2.2, it follows that either b \in C, q \in C and (a+ b)2 + p \in C; or
a \in C, p \in C and (a+ b)2 + q \in C.
Proposition 2.3 is proved.
Now we extend the previous results to the general case: At first we need to recall the following
notation: if f(x1, . . . , xn) is a multilinear polynomial over C, then we write
f(x1, . . . , xn) = x1x2 . . . xn +
\sum
\sigma \in Sn
\alpha \sigma x\sigma (1) . . . x\sigma (n)
for some \alpha \sigma \in C. Moreover, if d is a derivation of R, we denote by fd(x1, . . . , xn) the poly-
nomial obtained from f(x1, . . . , xn) by replacing each coefficient \alpha \sigma with d(\alpha \sigma ). Thus we write
d
\bigl(
f(r1, . . . , rn)
\bigr)
= fd(r1, . . . , rn) +
\sum
i
f
\bigl(
r1, . . . , d(ri), . . . , rn
\bigr)
, for all r1, r2, . . . , rn in R. We
also permit the following:
Remark 2.1 (Theorem 3 in [17]). Every generalized derivation g on a dense right ideal of R can
be uniquely extended to U and assumes the form g(x) = ax+d(x), for some a \in U and a derivation
d on U.
Fact 2.3. Let R be a prime K-algebra of characteristic different from 2 and f(x1, . . . , xn) a
multilinear polynomial over K. If for any i = 1, . . . , n,\bigl[
f(r1, . . . , zi, . . . , rn), f(r1, . . . , rn)
\bigr]
\in Z(R)
for all zi, r1, . . . , rn \in R, then the polynomial f(x1, . . . , xn) is central-valued on R.
Proof. Let s \in R. Then by assumption\bigl[
s, f(r1, . . . , rn)
\bigr]
2
=
=
\Biggl[ \sum
i
f(r1, . . . , [s, ri], . . . , rn), f(r1, . . . , rn)
\Biggr]
\in Z(R).
Hence, [s, f(r1, . . . , rn)]3 = [[s, f(r1, . . . , rn)]2, f(r1, . . . , rn)] = 0 and the result follows by [15]
(Theorem).
Fact 2.3 is proved.
As a reduction of the result in [6] we get:
Fact 2.4. Let R be a prime ring of characteristic different from 2, G a non-zero generalized
derivation of R and f(x1, . . . , xn) a multilinear polynomial over C. If\bigl[
G
\bigl(
f(x1, . . . , xn)
\bigr)
, f(x1, . . . , xn)
\bigr]
= 0
for all x1, . . . , xn \in R, then either there exists \alpha \in C such that G(x) = \alpha x or one of the following
holds:
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GENERALIZED DERIVATIONS AND COMMUTING ADDITIVE MAPS ON MULTILINEAR . . . 191
(1) f(x1, . . . , xn) is central valued in R;
(2) G(x) = cx+ xb with c - b \in C, and f(x1, . . . , xn)
2 is central valued in R.
In all that follows we denote by d and \delta the derivations of U such that F (x) = ax + d(x) and
G(x) = cx + \delta (x), for some a, c \in U and for all x \in R. We would like to permit the following
remark.
Remark 2.1. If F = 0 in Theorem 1.1, then either f(x1, . . . , xn) is central valued in R or the
particular cases of conclusions 1, 2, 4 in Theorem 1.1 are obtained.
In this case F 2 +G = G and then by Fact 2.4 either f(x1, . . . , xn) is central valued in R or one
of the following holds:
(1) there exists \alpha \in C such that G(x) = \alpha x for all x \in R. Hence, by easy calculations c = \alpha
and \delta = 0 (particular case of conclusions 1 and 2);
(2) there exist p, q \in U such that G(x) = px + xq for all x \in R with p - q \in C. Moreover,
f(x1, . . . , xn)
2 is central valued in R (particular case of conclusion 4).
Remark 2.2. If d = 0 in Theorem 1.1, then either f(x1, . . . , xn) is central valued in R or we
obtain particular cases of conclusions 2, 4 in Theorem 1.1.
In this case F (x) = ax and (F 2 +G)(x) = (a2 + c)x+ \delta (x) for all x \in R. Therefore F 2 +G
is a generalized derivation of R and again by Fact 2.4 either f(x1, . . . , xn) is central valued in R or
one of the following holds:
(1) there exists \alpha \in C such that (F 2 + G)(x) = \alpha x for all x \in R. By calculations, it follows
a2 + c = \alpha and \delta = 0 (particular case of conclusion 2);
(2) there exist p, q \in U such that (F 2 + G)(x) = px + xq for all x \in R with p - q = \gamma \in C.
Moreover, f(x1, . . . , xn)2 is central valued in R. By calculations, it follows G(x) = (q - a2)x +
+ x(q + \gamma ) (particular case of conclusion 4).
Proof of Theorem 1.1. We denote by d and \delta the derivations of U such that F (x) = ax+ d(x)
and G(x) = cx + \delta (x), for some a, c \in U and for all x \in R. In light of Remarks 2.1 and 2.2, we
may assume in all follows that F \not = 0 and d \not = 0.
Let fd(x1, . . . , xn), f
d\delta (x1, . . . , xn) be the polynomials obtained from f(x1, . . . , xn) replacing
each coefficient \alpha \sigma with d(\alpha \sigma ) and \delta (d(\alpha \sigma )) respectively. Thus we have
d
\bigl(
f(r1, . . . , rn)
\bigr)
= fd(r1, . . . , rn) +
\sum
i
f
\bigl(
r1, . . . , d(ri), . . . , rn
\bigr)
and similarly for \delta (f(r1, . . . , rn)). Moreover,
d2
\bigl(
f(x1, . . . , xn)
\bigr)
=
= fd2(x1, . . . , xn) + 2
\sum
i
fd(x1, . . . , d(xi), . . . , xn)+
+
\sum
i
f(x1, . . . , d
2(xi), . . . , xn) +
\sum
i \not =j
f(x1, . . . , d(xi), . . . , d(xj), . . . , xn).
By Remark 2.1, we have that R satisfies the following:
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192 V. DE FILIPPIS, B. DHARA, G. SCUDO\Bigl[
a2f(x1, . . . , xn) + 2ad(f(x1, . . . , xn)) + d(a)f(x1, . . . , xn)+
+d2(f(x1, . . . , xn)) + cf(x1, . . . , xn) + \delta (f(x1, . . . , xn)), f(x1, . . . , xn)
\Bigr]
that is \Biggl[
a2f(x1, . . . , xn) + 2afd(x1, . . . , xn) +
+2a
\sum
i
f(x1, . . . , d(xi), . . . , xn) + d(a)f(x1, . . . , xn)+
+fd2(x1, . . . , xn) + 2
\sum
i
fd(x1, . . . , d(xi), . . . , xn)+
+
\sum
i
f(x1, . . . , d
2(xi), . . . , xn) +
\sum
i \not =j
f(x1, . . . , d(xi), . . . , d(xj), . . . , xn)+
+cf(x1, . . . , xn) + f \delta (x1, . . . , xn) +
\sum
i
f(x1, . . . , \delta (xi), . . . , xn), f(x1, . . . , xn)
\Biggr]
. (2.3)
Suppose first that both d and \delta are inner derivations of R, that is, there exist b, q \in U such
that d(x) = [b, x] and \delta (x) = [q, x] for all x \in R. In this case F (x) = (a + b)x + x( - b) and
G(x) = (c+ q)x+ x( - q) for all x \in R. Then by Proposition 2.3, one of the following holds:
(1) a+ b, c+ q \in C, a2 + c \in C and F (x) = xa, G(x) = xc;
(2) b, q \in C, a2 + c \in C and F (x) = ax, G(x) = cx;
(3) f(x1, . . . , xn)
2 is central valued on R, a+ b \in C and F (x) = xa with a2 - 2q - c \in C;
(4) f(x1, . . . , xn)
2 is central valued on R, b \in C and F (x) = ax with a2 + 2q + c \in C;
unless f(x1, . . . , xn) is central valued on R, as required.
To complete the proof, in all that follows we consider the case when at least one of either F or G
is not an inner generalized derivation of R, that is, \delta and d are not simultaneously inner derivations
of R. We prove that if f(x1, . . . , xn) is not central valued on R, then this assumption leads to a
number of contradictions.
Suppose first that \delta and d are linearly C-independent modulo Dint (the set of inner derivations
in U).
In case \delta = 0, by [13], (2.3) gives that R satisfies\biggl[
a2f(x1, . . . , xn) + 2afd(x1, . . . , xn)+
+2a
\sum
i
f(x1, . . . , yi, . . . , xn) + d(a)f(x1, . . . , xn)+
+fd2(x1, . . . , xn) + 2
\sum
i
fd(x1, . . . , yi, . . . , xn)+
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GENERALIZED DERIVATIONS AND COMMUTING ADDITIVE MAPS ON MULTILINEAR . . . 193
+
\sum
i
f(x1, . . . , zi, . . . , xn) +
\sum
i \not =j
f(x1, . . . , yi, . . . , yj , . . . , xn)+
+cf(x1, . . . , xn), f(x1, . . . , xn)
\biggr]
.
On the other hand, if \delta \not = 0, again by [13], (2.3) gives that R satisfies
\biggl[
a2f(x1, . . . , xn) + 2afd(x1, . . . , xn)+
+2a
\sum
i
f(x1, . . . , yi, . . . , xn) + d(a)f(x1, . . . , xn)+
+fd2(x1, . . . , xn) + 2
\sum
i
fd(x1, . . . , yi, . . . , xn)+
+
\sum
i
f(x1, . . . , zi, . . . , xn) +
\sum
i \not =j
f(x1, . . . , yi, . . . , yj , . . . , xn)+
+cf(x1, . . . , xn) + f \delta (x1, . . . , xn) +
\sum
i
f(x1, . . . , ti, . . . , xn), f(x1, . . . , xn)
\biggr]
.
Notice that in both cases R satisfies the blended component\Bigl[
f(x1, . . . , zi, . . . , xn), f(x1, . . . , xn)
\Bigr]
for all i = 1, . . . , n. In light of Fact 2.3, this leads to the contradiction that f(x1, . . . , xn) is central
valued on R.
Suppose next that \delta and d are linearly C-dependent modulo Dint, that is, there exist \alpha , \beta \in C
and q \in U such that \alpha d+ \beta \delta = ad(q), the inner derivation induced by q
\bigl(
that is ad(q) = [q, x] for
all x \in R
\bigr)
. We divide this case into 3 subcases:
Case 1: \alpha = 0. In this case \delta (x) = [p, x] for all x \in R, with p = \beta - 1q. Moreover, d is not an
inner derivation.
Since d \not = 0, by [13], (2.3) gives that R satisfies\Bigl[
a2f(x1, . . . , xn) + 2afd(x1, . . . , xn)+
+2a
\sum
i
f(x1, . . . , yi, . . . , xn) + d(a)f(x1, . . . , xn)+
+fd2(x1, . . . , xn) + 2
\sum
i
fd(x1, . . . , yi, . . . , xn)+
+
\sum
i
f(x1, . . . , zi, . . . , xn) +
\sum
i \not =j
f(x1, . . . , yi, . . . , yj , . . . , xn)+
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194 V. DE FILIPPIS, B. DHARA, G. SCUDO
+ cf(x1, . . . , xn) + [p, f(x1, . . . , xn)], f(x1, . . . , xn)
\Bigr]
.
In particular R satisfies the component\Bigl[
f(x1, . . . , zi, . . . , xn), f(x1, . . . , xn)
\Bigr]
for all i = 1, . . . , n and then as above we get a contradiction.
Case 2: \beta = 0. In this case d = [p, x] for all x \in R, with p = \alpha - 1q /\in C. Moreover \delta is not an
inner derivation. Notice that in case \delta = 0 then both F and G are inner generalized derivations of R,
a contradiction. Thus \delta \not = 0. Then by [13], (2.3) gives that R satisfies\Biggl[
a2f(x1, . . . , xn) + 2a[p, f(x1, . . . , xn)] + d(a)f(x1, . . . , xn) +
+
\bigl[
p, [p, f(x1, . . . , xn)]
\bigr]
+
+cf(x1, . . . , xn) + f \delta (x1, . . . , xn)+
+
\sum
i
f(x1, . . . , yi, . . . , xn), f(x1, . . . , xn)
\Biggr]
.
In particular R satisfies \Bigl[
f(x1, . . . , yi, . . . , xn), f(x1, . . . , xn)
\Bigr]
for all i = 1, . . . , n, again leading a contradiction.
Case 3: \alpha \not = 0 and \beta \not = 0. In this case \delta = \gamma d+ ad(p), where \gamma = - \alpha \beta - 1 \not = 0 and ad(p) is the
inner derivation induced by the element p = \beta - 1q, moreover d is not an inner derivation of R.
Also here we notice that, in case \delta = 0 then both F and G are inner generalized derivations of R,
a contradiction. Thus \delta \not = 0 and by equation (2.3), we have that R satisfies\Biggl[
a2f(x1, . . . , xn) + 2afd(x1, . . . , xn) +
+2a
\sum
i
f(x1, . . . , d(xi), . . . , xn) + d(a)f(x1, . . . , xn)+
+fd2(x1, . . . , xn) + 2
\sum
i
fd(x1, . . . , d(xi), . . . , xn)+
+
\sum
i
f(x1, . . . , d
2(xi), . . . , xn) +
\sum
i \not =j
f(x1, . . . , d(xi), . . . , d(xj), . . . , xn)+
+cf(x1, . . . , xn) + \gamma fd(x1, . . . , xn)+
+\gamma
\sum
i
f(x1, . . . , d(xi), . . . , xn) + [p, f(x1, . . . , xn)], f(x1, . . . , xn)
\Biggr]
.
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GENERALIZED DERIVATIONS AND COMMUTING ADDITIVE MAPS ON MULTILINEAR . . . 195
Since d is not an inner derivation, by [13], from above relation, R satisfies\biggl[
a2f(x1, . . . , xn) + 2afd(x1, . . . , xn)+
+2a
\sum
i
f(x1, . . . , ui, . . . , xn) + d(a)f(x1, . . . , xn)+
+fd2(x1, . . . , xn) + 2
\sum
i
fd(x1, . . . , yi, . . . , xn)+
+
\sum
i
f(x1, . . . , zi, . . . , xn) +
\sum
i \not =j
f(x1, . . . , yi, . . . , yj , . . . , xn)+
+cf(x1, . . . , xn) + \gamma fd(x1, . . . , xn)+
+\gamma
\sum
i
f(x1, . . . , yi, . . . , xn) +
\bigl[
p, f(x1, . . . , xn)
\bigr]
, f(x1, . . . , xn)
\biggr]
.
In particular, for all i = 1, . . . , n, R satisfies\Bigl[
f(x1, . . . , zi, . . . , xn), f(x1, . . . , xn)
\Bigr]
.
Then again by Fact 2.3, we have a contradiction.
3. The case: pair of derivations on multilinear polynomials in right ideals. We would like
to point out the following reduced version of Theorem 1.1.
Theorem 3.1. Let R be a prime ring of characteristic different from 2, U its right Utumi
quotient ring, C its extended centroid, F and G two derivations of R, f(x1, . . . , xn) a multilinear
polynomial over C. If
\bigl[
(F 2 + G)
\bigl(
f(r1, . . . , rn)
\bigr)
, f(r1, . . . , rn)
\bigr]
= 0, for all r1, . . . , rn \in R, then
either F = G = 0 or f(x1, . . . , xn) is central valued on R.
To prove Theorem 1.2, we begin with the following remark.
Remark 3.1. Let R be a prime ring, f(x1, . . . , xn) a multilinear polynomial over C and I a
nonzero right ideal of R. By [18], following statements hold:
(1) if f(x1, . . . , xn)xn+1 is an identity for I, then there exists an idempotent element e \in
\in \mathrm{s}\mathrm{o}\mathrm{c}(RC) such that IC = eRC and f(x1, . . . , xn) is an identity for eRCe, so that a fortiori
f(x1, . . . , xn) is central valued in eRCe;
(2) if
\bigl[
f(x1, . . . , xn), xn+1
\bigr]
xn+2 is an identity for I, then there exists e2 = e \in \mathrm{s}\mathrm{o}\mathrm{c}(RC) such
that IC = eRC and f(x1, . . . , xn) is central valued in eRCe.
In light of Lemma 2.1, we have the following lemma.
Lemma 3.1. Let R be a prime ring, F (x) = cx - xc and G(x) = px - xp for c, p \in U be two in-
ner derivations of R. Let I be a right ideal of R such that
\bigl[
(F 2+G)
\bigl(
f(r1, . . . , rn)
\bigr)
, f(r1, . . . , rn)
\bigr]
=
= 0 for all r1, . . . , rn \in I. Then either R satisfies a nontrivial generalized polynomial identity or
there exist \alpha , \beta \in C such that (c - \alpha )I = (0), (p - \beta )I = (0) and (c - \alpha )2 = (p - \beta ).
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196 V. DE FILIPPIS, B. DHARA, G. SCUDO
Remark 3.2. We prefer to write the polynomial f(x1, . . . , xn) as follows:
f(x1, . . . , xn) =
\sum
i
gi(x1, . . . , xi - 1, xi+1, . . . , xn)xi,
where gi is a multilinear polynomial such that xi never appears in any monomials of gi. Note that
if there exists an idempotent e \in H = \mathrm{S}\mathrm{o}\mathrm{c}(RC) such that all gis are the polynomial identities
for eHe, then we get the conclusion that f(x1, . . . , xn) is a polynomial identity for eHe. Thus if
f(x1, . . . , xn) is not a polynomial identity for eHe, there exists an index i and r1, . . . , rn - 1 \in eHe
such that gi(r1, . . . , rn - 1) \not = 0. Without loss of generality we assume i = n, say gn(x1, . . . , xn - 1) =
= t(x1, . . . , xn - 1) and so f(x1, . . . , xn) = t(x1, . . . , xn - 1)xn + h(x1, . . . , xn) where t(eHe) \not = 0
and h(x1, . . . , xn) is a multilinear polynomial such that xn never appears as last variable in any
monomials of h.
Lemma 3.2. Let R be a prime ring of characteristic different from 2, U its right Utumi
quotient ring, C its extended centroid, F and G two inner derivations of R induced by the
elements a, b \in U respectively, that is F (x) = [a, x] and G(x) = [b, x] for all x \in R. If
f(x1, . . . , xn) is a multilinear polynomial over C, I a non-zero right ideal of R such that
\bigl[
(F 2 +
+G)
\bigl(
f(r1, . . . , rn)
\bigr)
, f(r1, . . . , rn)
\bigr]
= 0 for all r1, . . . , rn \in I, then one of the following holds:
(1) there exists e2 = e \in soc(RC) such that IC = eRC and f(x1, . . . , xn) is central valued on
eRCe;
(2) there exist \alpha , \beta \in C such that (a - \alpha )I = (b - \beta )I = (0) and (a - \alpha )2 = (b - \beta ).
Proof. By Lemma 3.1, we may assume that R satisfies a nontrivial generalized polynomial
identity, otherwise we get our conclusion (2). In this case by [23], RC is a primitive ring having a
non-zero socle H with a non-zero right ideal J = IH. Note that H is simple, J = HJ and J satisfies
the same basic conditions as I. Thus without loss of generality we may replace R by H and I by J.
Since R = H is a regular ring, then for any a1, . . . , an \in I there exists h = h2 \in R such that\sum n
i=1
aiR = hR. Then h \in IR = I and ai = hai for each i = 1, . . . , n.
By our assumption, I satisfies the following generalized identity with coefficients in U :\bigl[
(a2 + b)f(x1, . . . , xn) - 2af(x1, . . . , xn)a+
+f(x1, . . . , xn)(a
2 - b), f(x1, . . . , xn)
\bigr]
. (3.1)
First we study the situation when there exists \alpha \in C such that (a - \alpha )I = (0). Notice that a and
c = a - \alpha induce the same derivation F. Thus we replace a by c and assume cI = (0).
By calculations, (3.1) reduces to\Bigl[
bf(x1, . . . , xn) + f(x1, . . . , xn)(c
2 - b), f(x1, . . . , xn)
\Bigr]
. (3.2)
By Theorem 3 in [6], we have from (3.2) that either there exists e = e2 \in \mathrm{S}\mathrm{o}\mathrm{c}(RC) such that
IC = eRC and f(x1, . . . , xn) is central valued in eRCe or one of the following holds:
1. There exists \beta \in C such that (b - \beta )I = (0) and c2 - b \in C. Applying this to (3.1), it follows
that I satisfies
f(x1, . . . , xn)
2
\bigl(
(a - \alpha )2 - (b - \beta )
\bigr)
= 0
and by the main result in [5], we get the required conclusion (a - \alpha )2 = (b - \beta ), unless when there
exists e = e2 \in \mathrm{S}\mathrm{o}\mathrm{c}(RC) such that IC = eRC and f(x1, . . . , xn)xn+1 is an identity for eRC.
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GENERALIZED DERIVATIONS AND COMMUTING ADDITIVE MAPS ON MULTILINEAR . . . 197
2. There exists \beta \in C such that (c2 - 2b - \beta )I = (0), that is (b + \gamma )I = (0), with \gamma =
\beta
2
;
moreover there exists e = e2 \in \mathrm{S}\mathrm{o}\mathrm{c}(RC) such that IC = eRC and f(x1, . . . , xn)
2 is central valued
in eRCe. Since eRCe satisfies (3.1), also in this case, by calculations we have that eRCe satisfies
f(x1, . . . , xn)
2
\bigl(
c2 - (b+ \gamma )
\bigr)
= 0,
that is (a - \alpha )2 = (b+ \gamma ). In any case we obtain one of the required conclusions.
Therefore, in what follows we may assume that there exist c, c1, . . . , cn+2 \in I such that ac \not = \alpha c
for all \alpha \in C and [f(c1, . . . , cn), cn+1]cn+2 \not = 0.
By the above argument, there exists an idempotent element e \in IH = IR such that eR =
=
\sum n+2
i=1
ciR + cR + aR + bR and ci = eci (for any i = 1, . . . , n + 2), c = ec, a = ea, b = eb.
Notice that \Bigl[
(a2 + b)f(ex1, . . . , exn) - 2af(ex1, . . . , exn)a+
+f(ex1, . . . , exn)(a
2 - b), f(ex1, . . . , exn)
\Bigr]
(3.3)
is satisfied by R = H. Now we write the polynomial f(x1, . . . , xn) as in Remark 3.2, and replace
xn by xn(1 - e). Hence
f
\bigl(
ex1, . . . , exn - 1, exn(1 - e)
\bigr)
= t(ex1, . . . , exn - 1)exn(1 - e). (3.4)
By using (3.4) in (3.3) and right multiplying by e, we have that R satisfies
2t(ex1, . . . , exn - 1)exn(1 - e)at(ex1, . . . , exn - 1)exn(1 - e)ae
and since \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(R) \not = 2 and t(ex1, . . . , exn)e \not = 0, it follows (1 - e)ae = 0, that is ae = eae and
a2e = aeae = eaeae.
In light of this, R satisfies\Bigl[
e(a2 + b)ef(ex1e, . . . , exne) - 2eaef(ex1e, . . . , exne)eae+
+f(ex1e, . . . , exne)e(a
2 - b)e, f(ex1e, . . . , exne)
\Bigr]
that is eRCe satisfies\biggl[ \bigl(
e(a2 + b)e
\bigr)
f(x1, . . . , xn) - 2(eae)f(x1, . . . , xn)(eae)+
+f(x1, . . . , xn)
\bigl(
e(a2 - b)e
\bigr)
, f(x1, . . . , xn)
\biggr]
.
Since eRCe is a is a simple ring, by Theorem 3.1, we have that both eae \in Z(eRCe) and ebe \in
\in Z(eRCe), since f(x1, . . . , xn) is not central valued on eRCe. In particular there exists \alpha \in C
such that \alpha e = eae = ae, that is \alpha c = \alpha ec = aec = ac, which is a contradiction.
Lemma 3.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2
198 V. DE FILIPPIS, B. DHARA, G. SCUDO
We are finally ready for the proof of Theorem 1.2.
Proof of Theorem 1.2. Of course we may assume that F and G are not simultaneously inner
derivations of R, if not we end up by Lemma 3.2. Moreover in case either F = 0 or G = 0, the
conclusion follows respectively from [16] (see also [6]) and [9]. Therefore we always assume that
F \not = 0 and G \not = 0.
By our hypothesis, if 0 \not = c \in I, R satisfies\Bigl[
F 2(f(cx1, . . . , cxn)) +G(f(cx1, . . . , cxn)), f(cx1, . . . , cxn)
\Bigr]
,
that is R satisfies\Biggl[
fF 2
(cx1, . . . , cxn) + 2
\sum
i
fF (cx1, . . . , F (c)xi + cF (xi), . . . , cxn)+
+
\sum
i
f(cx1, . . . , F
2(c)xi + 2F (c)F (xi) + cF 2(xi), . . . , cxn)+
+
\sum
i \not =j
f(cx1, . . . , F (c)xi + cF (xi), . . . , F (c)xj + cF (xj), . . . , cxn)+
+fG(cx1, . . . , cxn) +
\sum
i
f(cx1, . . . , G(c)xi + cG(xi), . . . , cxn), f(cx1, . . . , cxn)
\Biggr]
. (3.5)
In all that follows we consider the case when at least one of either F or G is not an inner
derivation of R. Moreover, we assume that there exist c1, . . . , cn+2 \in I such that\bigl[
f(c1, . . . , cn), cn+1
\bigr]
cn+2 \not = 0,
otherwise by Remark 3.1, we obtain conclusion (1).
Suppose first that F and G are linearly C-independent modulo Dint.
By [13], we have from (3.5) that R satisfies\Biggl[
fF 2
(cx1, . . . , cxn) + 2
\sum
i
fF (cx1, . . . , F (c)xi + cyi, . . . , cxn)+
+
\sum
i
f(cx1, . . . , F
2(c)xi + 2F (c)yi + czi, . . . , cxn)+
+
\sum
i \not =j
f(cx1, . . . , F (c)xi + cyi, . . . , F (c)xj + cyj , . . . , cxn)+
+fG(cx1, . . . , cxn) +
\sum
i
f(cx1, . . . , G(c)xi + cti, . . . , cxn), f(cx1, . . . , cxn)
\Biggr]
.
This implies that R satisfies the blended component
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2
GENERALIZED DERIVATIONS AND COMMUTING ADDITIVE MAPS ON MULTILINEAR . . . 199\Biggl[ \sum
i
f(cx1, . . . , cti, . . . , cxn), f(cx1, . . . , cxn)
\Biggr]
.
Suppose next that F and G are linearly C-dependent modulo Dint, that is, there exist \alpha , \beta \in C
and q \in U such that \alpha F +\beta G = ad(q), the inner derivation induced by q. We divide this case into 3
subcases:
Case 1: \alpha = 0. In this case G(x) = [p, x] for all x \in R, with p = \beta - 1q, moreover F is not an
inner derivation. By (3.5), we have that R satisfies\Biggl[
fF 2
(cx1, . . . , cxn) + 2
\sum
i
fF (cx1, . . . , F (c)xi + cyi, . . . , cxn)+
+
\sum
i
f(cx1, . . . , F
2(c)xi + 2F (c)yi + czi, . . . , cxn)+
+
\sum
i \not =j
f(cx1, . . . , F (c)xi + cyi, . . . , F (c)xj + cyj , . . . , cxn)+
+ pf(cx1, . . . , cxn) - f(cx1, . . . , cxn)p, f(cx1, . . . , cxn)
\Biggr]
.
In particular, R satisfies the component\Bigl[
f(cx1, . . . , czi, . . . , cxn), f(cx1, . . . , cxn)
\Bigr]
for all i = 1, . . . , n.
Case 2: \beta = 0. In this case F = [p, x] for all x \in R, with p = \alpha - 1q, moreover G is not an inner
derivation. By (3.5), we have that R satisfies\Biggl[
p2f(cx1, . . . , cxn) - 2pf(cx1, . . . , cxn)p+ f(cx1, . . . , cxn)p
2 +
+fG(cx1, . . . , cxn) +
\sum
i
f(cx1, . . . , G(c)xi + cyi, . . . , cxn), f(cx1, . . . , cxn)
\Biggr]
.
In particular \Bigl[
f(cx1, . . . , cyi, . . . , cxn), f(cx1, . . . , cxn)
\Bigr]
is satisfied by R, for all i = 1, . . . , n.
Case 3: \alpha \not = 0 and \beta \not = 0. In this case G = \gamma F + ad(p), where \gamma = - \alpha \beta - 1 \not = 0 and ad(p) is
the inner derivation induced by the element p = \beta - 1q, moreover F is not an inner derivation of R.
By equation (3.5), we have that R satisfies
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2
200 V. DE FILIPPIS, B. DHARA, G. SCUDO\Biggl[
fF 2
(cx1, . . . , cxn) + 2
\sum
i
fF (cx1, . . . , F (c)xi + cyi, . . . , cxn)+
+
\sum
i
f(cx1, . . . , F
2(c)xi + 2F (c)yi + czi, . . . , cxn)+
+
\sum
i \not =j
f(cx1, . . . , F (c)xi + cyi, . . . , F (c)xj + cyj , . . . , cxn)+
+\gamma fF (cx1, . . . , cxn)+
+ \gamma
\sum
i
f(cx1, . . . , F (c)xi + cyi, . . . , cxn) + [p, f(cx1, . . . , cxn)], f(cx1, . . . , cxn)
\Biggr]
.
In particular, for all i = 1, . . . , n, R satisfies\Bigl[
f(cx1, . . . , czi, . . . , cxn), f(cx1, . . . , cxn)
\Bigr]
. (3.6)
All the previous argument says that in any case R satisfies (3.6). Thus R satisfies a nontrivial
generalized polynomial identity. As remarked in the proof of Lemma 3.2, we may assume H = R
and I = IR. Moreover, for all e2 = e \in I and by the above argument, R satisfies\Biggl[ \sum
i
f(ex1, . . . , eti, . . . , exn), f(ex1, . . . , exn)
\Biggr]
. (3.7)
Since R = H is a regular ring, then there exists h = h2 \in R such that
\sum n+2
i=1
ciR = hR. Then
h \in IR = I and ci = hci for each i = 1, . . . , n+ 2. In (3.7) and for all i = 1, . . . , n, we replace hti
by [hcn+1, hxi], so that R satisfies \bigl[
cn+1, f(hx1, . . . , hxn)
\bigr]
2
.
In particular the ring hRh satisfies
\bigl[
cn+1, f(x1, . . . , xn)
\bigr]
2
. By [15], it follows cn+1 \in Z(hRh), and
a fortiori
\bigl[
cn+1, f(c1, . . . , cn)
\bigr]
cn+2 = 0, which is a contradiction.
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Received 25.04.13
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2
|
| id | umjimathkievua-article-1833 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:13:30Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/03/636859742cff9ee3216d1fbd328d9403.pdf |
| spelling | umjimathkievua-article-18332019-12-05T09:29:16Z Generalized derivations and commuting additive maps on multilinear polynomials in prime rings Узагальненi похiднi та комутуючi адитивнi вiдображення на мультилiнiйних полiномах у простих кiльцях De, Filippis V. Dhara, B. Scudo, G. Де, Філіппіс В. Дхара, B. Сцудо, Г. Let $R$ be a prime ring with characteristic different from $2, U$ be its right Utumi quotient ring, $C$ be its extended centroid, $F$ and $G$ be additive maps on $R$ , $f(x_1, ..., x_n)$ be a multilinear polynomial over $C$, and $I$ be a nonzero right ideal of $R$ . We obtain information about the structure of $R$ and describe the form of $F$ and $G$ in the following cases: $$(1) [(F^2 + G)(f(r_1, ..., r_n)), f(r_1, ..., r_n)] = 0$$ for all $r_1, . . . , r_n \in R$, where $F$ and $G$ are generalized derivations of $R$ ; $$(2) [(F^2 + G)(f(r_1, ..., r_n)), f(r_1, ..., r_n)] = 0$$for all $r_1, ..., r_n \in I$, where $F$ and $G$ are derivations of $R$. Нехай $R$ — просте кiльце з характеристикою, що вiдмiнна вiд $2, U$ — його праве фактор-кiльце, $C$ — його розширений центроїд, $F$ та $G$ — адитивнi вiдображення на $R, f(x_1, ..., x_n)$ — мультилiнiйний полiном над $C$, а $I$ — ненульовий правий iдеал для $R$. Отримано iнформацiю про структуру кiльця $R$ та описано форму $F$ i $G$ у таких випадках: $$(1) [(F^2 + G)(f(r_1, ..., r_n)), f(r_1, ..., r_n)] = 0$$ для всiх $r_1, . . . , r_n \in R$, де $F$ та $G$ — узагальненi похiднi вiд $R$ ; $$(2) [(F^2 + G)(f(r_1, ..., r_n)), f(r_1, ..., r_n)] = 0$$ для всiх $r_1, ..., r_n \in I$, де $F$ та $G$ — похiднi вiд $R$ . Institute of Mathematics, NAS of Ukraine 2016-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1833 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 2 (2016); 183-201 Український математичний журнал; Том 68 № 2 (2016); 183-201 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1833/815 Copyright (c) 2016 De Filippis V.; Dhara B.; Scudo G. |
| spellingShingle | De, Filippis V. Dhara, B. Scudo, G. Де, Філіппіс В. Дхара, B. Сцудо, Г. Generalized derivations and commuting additive maps on multilinear polynomials in prime rings |
| title | Generalized derivations and commuting additive maps
on multilinear polynomials in prime rings |
| title_alt | Узагальненi похiднi та комутуючi адитивнi вiдображення на мультилiнiйних полiномах у простих кiльцях |
| title_full | Generalized derivations and commuting additive maps
on multilinear polynomials in prime rings |
| title_fullStr | Generalized derivations and commuting additive maps
on multilinear polynomials in prime rings |
| title_full_unstemmed | Generalized derivations and commuting additive maps
on multilinear polynomials in prime rings |
| title_short | Generalized derivations and commuting additive maps
on multilinear polynomials in prime rings |
| title_sort | generalized derivations and commuting additive maps
on multilinear polynomials in prime rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1833 |
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