Generalized derivations acting on multilinear polynomials as a Jordan homomorphisms
UDC 512.5 Let $R$ be a prime ring whose characteristic is not equal to $2,$ let  $U$ be the Utumi quotient ring of $R,$ and let $C$ be the extended centroid of $R.$  Also let $G$ and $H$ be two generalized derivations on $R$ and let $f(x_1,\ldots,x...
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| author | Tiwari, S. K. Prajapati, B. Tiwari, S. K. Prajapati, B. |
| author_facet | Tiwari, S. K. Prajapati, B. Tiwari, S. K. Prajapati, B. |
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| description | UDC 512.5
Let $R$ be a prime ring whose characteristic is not equal to $2,$ let  $U$ be the Utumi quotient ring of $R,$ and let $C$ be the extended centroid of $R.$  Also let $G$ and $H$ be two generalized derivations on $R$ and let $f(x_1,\ldots,x_n)$ be a noncentral multilinear polynomial over $C.$  If $G(H(u^2))=(H(u))^2$ for all $u=f(r_1,\ldots,r_n),$ $r_1,\ldots,r_n \in R,$ then one of the following holds:
1) $H=0;$
2) there exists $\lambda\in C$ such that $G(x)=H(x)=\lambda x$ for all $x\in R;$
3) there exist $\lambda\in C$ and $a\in U$ such that $H(x)=\lambda x$ and $G(x)=[a, x]+\lambda x$ for all $x\in R$ and $f(x_1,\ldots,x_n)^2$ is central-valued on $R.$ |
| doi_str_mv | 10.37863/umzh.v74i7.6108 |
| first_indexed | 2026-03-24T03:26:05Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i7.6108
UDC 512.5
S. K. Tiwari (Indian Institute of Technology Patna, Bihar, India),
B. Prajapati (School of Liberal Studies, Ambedkar University Delhi, India)
GENERALIZED DERIVATIONS ACTING ON MULTILINEAR POLYNOMIALS
AS A JORDAN HOMOMORPHISMS
УЗАГАЛЬНЕНI ПОХIДНI, ЩО ДIЮТЬ НА МУЛЬТИЛIНIЙНИХ ПОЛIНОМАХ
ЯК ЖОРДАНОВI ГОМОМОРФIЗМИ
Let R be a prime ring whose characteristic is not equal to 2, let U be the Utumi quotient ring of R, and let C be the
extended centroid of R. Also let G and H be two generalized derivations on R and let f(x1, . . . , xn) be a noncentral
multilinear polynomial over C. If G(H(u2)) = (H(u))2 for all u = f(r1, . . . , rn), r1, . . . , rn \in R, then one of the
following holds:
1) H = 0;
2) there exists \lambda \in C such that G(x) = H(x) = \lambda x for all x \in R;
3) there exist \lambda \in C and a \in U such that H(x) = \lambda x and G(x) = [a, x] + \lambda x for all x \in R and f(x1, . . . , xn)
2 is
central-valued on R.
Нехай R — просте кiльце з характеристикою, що не дорiвнює 2, U — фактор-кiльце Утумi для R, а C — продовжений
центроїд для R. Крiм того, припустимо, що G та H — двi узагальненi похiднi на R, а f(x1, . . . , xn) — нецентраль-
ний мультилiнiйний полiном над C. Якщо G(H(u2)) = (H(u))2 для всiх u = f(r1, . . . , rn), r1, . . . , rn \in R, то
справджується одне з таких тверджень:
1) H = 0;
2) iснує таке \lambda \in C, що G(x) = H(x) = \lambda x для всiх x \in R;
3) iснують такi \lambda \in C та a \in U, що H(x) = \lambda x, G(x) = [a, x] + \lambda x для всiх x \in R i f(x1, . . . , xn)
2
є центральнозначним на R.
1. Introduction. Throughout the article R denotes a prime ring of characteristic different from 2
with center Z(R) and U denotes the Utumi quotient ring of R. The center of U denoted by C is
called the extended centroid of R. An additive mapping d on a ring R is said to be a derivation if
d(xy) = d(x)y + xd(y) for all x, y \in R. For a fixed a \in R, the mapping da : R \rightarrow R, defined by
da(x) = [a, x], for all x \in R is a derivation, usually called inner derivation induced by an element
a \in R. A derivation is called outer if it is not an inner derivation. An additive mapping H on a ring
R is said to be a generalized derivation associated with a derivation d if H(xy) = H(x)y + xd(y)
for all x, y \in R. For fixed a, a\prime \in R, the mapping F(a,a\prime ) : R \rightarrow R defined by F(a,a\prime )(x) = ax+ xa\prime
is a generalized derivation on R. The mapping F(a,a\prime ) is usually called generalized inner derivation
on R.
An additive mapping on a ring R is a homomorphism if H(xy) = H(x)H(y) for all x, y \in R
and H is said to an anti-homomorphism if H(xy) = H(y)H(x) for all x, y \in R. An additive
mapping H is said to be a Jordan homomorphism if H(x2) = (H(x))2 for all x \in R. We observe
that every homomorphism and anti-homomorphism is a Jordan homomorphism but the converse is
not true in general. Following example justify our observation.
Example 1.1. Suppose that \ast is an involution on ring R and S = R
\bigoplus
R is a ring such that
r1ar2 = 0 for all r1, r2 \in R, where a \in Z(R). Define a function \zeta on S such that \zeta (r1, r2) =
c\bigcirc S. K. TIWARI, B. PRAJAPATI, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7 991
992 S. K. TIWARI, B. PRAJAPATI
= (ar1, r
\ast
2) for all r1, r2 \in R. This example shows that \zeta is a Jordan homomorphism but not a
homomorphism.
Herstein [13], in 1956 proved that every Jordan homomorphism from a ring R onto a prime ring
R\prime with char(R) \not = 2, 3 is either a homomorphism or anti-homomorphism. Further, Smiley [20], in
1957 improve the above result by removing the restriction of characteristic is not equal to 3 in the
hypothesis of the Herstein’s [13].
The context of derivation, which acts as a homomorphism or as an anti-homomorphism, was first
studied by Bell and Kappe [8]. More precisely, they proved that there is no nonzero derivation on
prime ring which acts as a homomorphism or as an anti-homomorphism on right ideal of R. Later
on many mathematician have studied the additive mapping which acts as a homomorphism, anti-
homomorphism, Jordan homomorphism, Lie homomorphism on some subsets of a particular ring.
For more details, we refer to reader [1 – 6, 21 – 25].
Recently, in this line of investigation De Filippis and Dhara [4], in 2019 studied the structure
of prime ring R, when generalized skew derivation acts as a Jordan homomorphism on multilinear
polynomial over C.
Motivated by above cited results, we would like to study the following.
Theorem 1.1. Let R be a prime ring of characteristic not equal to 2, U be the Utumi quotient
ring of R and C be the extended centroid of R. Let G and H be two generalized derivations
on R and f(x1, . . . , xn) be a multilinear polynomial over C which is noncentral valued on R. If
G(H(u2)) = (H(u))2 for all u = f(r1, . . . , rn), r1, . . . , rn \in R, then one of the following holds:
1) H = 0;
2) there exists \lambda \in C such that G(x) = H(x) = \lambda x for all x \in R;
3) there exist \lambda \in C and a \in U such that H(x) = \lambda x, G(x) = [a, x] + \lambda x for all x \in R and
f(x1, . . . , xn)
2 is central valued on R.
The following corollaries are an immediate application of Theorem 1.1. In particular, for G = I,
identity mapping in Theorem 1.1, we have the following.
Corollary 1.1. Let R be a prime ring of characteristic not equal to 2, U be the Utumi quotient
ring of R and C be the extended centroid of R. Let H be a nonzero generalized derivation on
R and f(x1, . . . , xn) be a multilinear polynomial over C which is noncentral valued on R. If
H(u2) = (H(u))2 for all u = f(r1, . . . , rn), r1, . . . , rn \in R, then H(x) = x for all x \in R.
In particular, for H = I, the identity mapping on R in Theorem 1.1, we have the following.
Corollary 1.2. Let R be a prime ring of characteristic not equal to 2, U be the Utumi quotient
ring of R and C be the extended centroid of R. Let H be a nonzero generalized derivation on R and
f(x1, . . . , xn) be a multilinear polynomial over C which is noncentral valued on R. If H(u2) = u2
for all u = f(r1, . . . , rn), r1, . . . , rn \in R, then one of the following holds:
1) H(x) = x for all x \in R;
2) there exists a \in U such that H(x) = [a, x] + x for all x \in R and f(x1, . . . , xn)
2 is central
valued on R.
Corollary 1.3. Let R be a prime ring of characteristic not equal to 2, U be the Utumi quotient
ring of R and C be the extended centroid of R. Let q /\in Z(R), H be a generalized derivation on R
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
GENERALIZED DERIVATIONS ACTING ON MULTILINEAR POLYNOMIALS . . . 993
and f(x1, . . . , xn) be a multilinear polynomial over C which is noncentral valued on R such that
[q,H(u2)] = (H(u))2 for all u = f(r1, . . . , rn), r1, . . . , rn \in R. Then H = 0.
2. Notations and known results. Let d and \delta be two derivations on R. We denote by
fd(x1, . . . , xn) the polynomials obtained from f(x1, . . . , xn) replacing each coefficients \alpha \sigma with
d(\alpha \sigma ). Then we have
d(f(x1, . . . , xn)) = fd(x1, . . . , xn) +
\sum
i
f(x1, . . . , d(xi), . . . , xn)
and
d\delta (f(r1, . . . , rn)) = fd\delta (r1, . . . , rn) +
\sum
i
fd(r1, . . . , \delta (ri), . . . , rn) +
+
\sum
i
f \delta (r1, . . . , d(ri), . . . , rn) +
\sum
i
f(r1, . . . , d\delta (ri), . . . , rn) +
+
\sum
i \not =j
f(r1, . . . , d(ri), . . . , \delta (rj), . . . , rn).
The following facts are frequently used to prove our results.
Fact 2.1. Let R be a prime ring and I a two-sided ideal of R. Then R, I and U satisfy the
same generalized polynomial identities with coefficients in U [10].
Fact 2.2. Let R be a prime ring and I a two-sided ideal of R. Then R, I and U satisfy the
same differential identities [17].
Fact 2.3. Let R be a prime ring. Then every derivation d of R can be uniquely extended to a
derivation of U (see Proposition 2.5.1 [7]).
Fact 2.4 ([15], Theorem 2). Let R be a prime ring, d a nonzero derivation on R and I a
nonzero ideal of R. If I satisfies the differential identity
f(r1, . . . , rn, d(r1), . . . , d(rn)) = 0
for any r1, . . . , rn \in I, then either
(i) I satisfies the generalized polynomial identity
f(r1, . . . , rn, x1, . . . , xn) = 0
or
(ii) d is U -inner, i.e., for some q \in U, d(x) = [q, x] and I satisfies the generalized polynomial
identity
f(r1, . . . , rn, [q, r1], . . . , [q, rn]) = 0.
Fact 2.5 ([6], Lemma 2.9). Let R be a prime ring of characteristic with \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(R) \not = 2, a, b, c,
c\prime \in U and p(x1, . . . , xn) be any polynomial over C which is not identity for R. If ap(r) + p(r)b+
+ cp(r)c\prime = 0 for all r \in Rn, then one of the following conditions holds:
1) b, c\prime \in C and a+ b+ cc\prime = 0;
2) a, c \in C and a+ b+ cc\prime = 0;
3) a+ b+ cc\prime = 0 and p(x1, . . . , xn)
2 is central valued on R.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
994 S. K. TIWARI, B. PRAJAPATI
3. \bfitG and \bfitH are generalized inner derivations. In this section, we study the case when G
and H are generalized inner derivations. Suppose that G(x) = ax+ xb and H(x) = px+ xq for all
x \in R and for some a, b, p, q \in U. From the given identity G(H(f(r)2)) = H(f(r))2 we get the
expression a\prime f(r)2+af(r)2q+pf(r)2b+f(r)2b\prime = pf(r)pf(r)+pf(r)2q+f(r)p\prime f(r)+f(r)qf(r)q
where a\prime = ap, b\prime = qb and p\prime = qp. To prove main result we prove the following propositions.
Proposition 3.1. Let R be a prime ring of characteristic not equal to 2, U be the Utumi quotient
ring of R and C be the extended centroid of R. Let G and H be two generalized inner derivations
on R and f(x1, . . . , xn) be a multilinear polynomial over C which is noncentral valued on R. If
G(H(u2)) = (H(u))2 for all u = f(r1, . . . , rn), r1, . . . , rn \in R, then one of the following holds:
1) H = 0;
2) there exists \lambda \in C such that G(x) = H(x) = \lambda x for all x \in R;
3) there exist \lambda \in C and a \in U such that H(x) = \lambda x, G(x) = [a, x] + \lambda x and f(x1, . . . , xn)
2
is central valued on R.
To prove the above proposition we need the following results.
Proposition 3.2. Let R = Mm(K) be the ring of all m \times m matrices over the field K with
characteristic not equal to 2 and m \geq 2 and f(x1, . . . , xn) be a noncentral multilinear polynomial
over K. Let a, a\prime , b, b\prime , p, p\prime , q \in U such that a\prime f(r)2+af(r)2q+pf(r)2b+f(r)2b\prime = pf(r)pf(r)+
+pf(r)2q+f(r)p\prime f(r)+f(r)qf(r)q for all r = (r1, . . . , rn) \in Rn. Then p \in K \cdot Im and q \in K \cdot Im.
Proof. Since f(x1, . . . , xn) be a noncentral on R. By [18] (Lemma 2, Proof of Lemma 3), there
exists a sequence of matrices r = (r1, . . . , rn) in R such that f(r1, . . . , rn) = \gamma eij with 0 \not = \gamma \in K
and i \not = j. Since the set f(R) = \{ f(x1, . . . , xn) | xi \in R\} is invariant under the action of all inner
automorphisms of R for all i \not = j there exists a sequence of matrices r = (r1, . . . , rn) in R such
that f(r1, . . . , rn) = \gamma eij . Thus our hypothesis
a\prime f(r1, . . . , rn)
2 + af(r1, . . . , rn)
2q + pf(r1, . . . , rn)
2b+ f(r1, . . . , rn)
2b\prime =
= pf(r1, . . . , rn)pf(r1, . . . , rn) + pf(r1, . . . , rn)
2q +
+ f(r1, . . . , rn)p
\prime f(r1, . . . , rn) + f(r1, . . . , rn)qf(r1, . . . , rn)q. (1)
Gives that
peijpeij + eijp
\prime eij + eijqeijq = 0.
Left multiplying above relation by eij , we obtain eijpeijpeij = 0. It implies that p2ij = 0 and hence
pij = 0 with i \not = j. It implies that p is a diagonal matrix.
Right multiplication by eij in above expression we get qij = 0 with i \not = j. This implies that q is
a diagonal matrix.
For any K -automorphism \theta of R, p\theta enjoy the same property as p does, p\theta must be diagonal.
Write p =
\sum m
i=1
piieii; then, for s \not = t, we have
(1 + ets)p(1 - ets) =
m\sum
i=1
piieii + (pss - ptt)ets
diagonal. Hence, pss = ptt and so p is a scalar matrix, that is, p \in K \cdot Im. Similarly, we can show
that q is diagonal and hence central.
Proposition 3.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
GENERALIZED DERIVATIONS ACTING ON MULTILINEAR POLYNOMIALS . . . 995
Lemma 3.1. Let R be a prime ring of characteristic not equal to 2. Let U be the Utumi ring
of quotients and C be the extended centroid of ring R. Suppose that f(x1, . . . , xn) be a multilinear
polynomial over C which is not central valued on R such that a\prime f(r)2 + af(r)2q + pf(r)2b +
+ f(r)2b\prime = pf(r)pf(r) + pf(r)2q + f(r)p\prime f(r) + f(r)qf(r)q for all r \in Rn and for some a, a\prime ,
b, b\prime , p, p\prime , q \in U. Then p and q are central.
Proof. On contrary suppose that both p and q are not central. By hypothesis, we have
\mathrm{h}(x1, . . . , xn) = a\prime f(x1, . . . , xn)
2 + af(x1, . . . , xn)
2q + pf(x1, . . . , xn)
2b +
+ f(x1, . . . , xn)
2b\prime - pf(x1, . . . , xn)pf(x1, . . . , xn) -
- pf(x1, . . . , xn)
2q - f(x1, . . . , xn)p
\prime f(x1, . . . , xn) -
- f(x1, . . . , xn)qf(x1, . . . , xn)q
for all x1, . . . , xn \in R, that is,
\mathrm{h}(x1, . . . , xn) = a\prime f(x1, . . . , xn)
2 +
\Bigl\{
af(x1, . . . , xn)
2 - pf(x1, . . . , xn)
2 -
- f(x1, . . . , xn)qf(x1, . . . , xn)
\Bigr\}
q + pf(x1, . . . , xn)
2b +
+ f(x1, . . . , xn)
2b\prime - pf(x1, . . . , xn)pf(x1, . . . , xn) -
- f(x1, . . . , xn)p
\prime f(x1, . . . , xn) (2)
for all x1, . . . , xn \in R.
Since R and U satisfy same generalized polynomial identity (GPI) (see [10]), U satisfies
\mathrm{h}(x1, . . . , xn) = 0T . Suppose that \mathrm{h}(x1, . . . , xn) is a trivial GPI for U. Let T = U\ast CC\{ x1, . . . , xn\} ,
the free product of U and C\{ x1, . . . , xn\} , the free C -algebra in non commuting indeterminates
x1, . . . , xn. Then \mathrm{h}(x1, . . . , xn) is zero element in T = U \ast C C\{ x1, . . . , xn\} . It implies that
\{ b, b\prime , q, 1\} is linearly C -dependent. Then there exist \alpha 1, \alpha 2, \alpha 3 and \alpha 4 \in C such that \alpha 1b +
+ \alpha 2b
\prime + \alpha 3q + \alpha 41 = 0. If \alpha 1 = 0 = \alpha 2, then \alpha 3 \not = 0 and so q = - \alpha - 1
3 \alpha 4 \in C, gives a
contradiction. Therefore either \alpha 1 \not = 0 or \alpha 2 \not = 0. Without loss of generality, we assume that
\alpha 1 \not = 0. Then b = \alpha b\prime + \beta q + \gamma , where \alpha = - \alpha - 1
1 \alpha 2, \beta = - \alpha - 1
1 \alpha 3 and \gamma = - \alpha - 1
1 \alpha 4. Then U
satisfies
(a\prime + p\gamma )f(x1, . . . , xn)
2 +
\Bigl\{
af(x1, . . . , xn)
2 - pf(x1, . . . , xn)
2 -
- f(x1, . . . , xn)qf(x1, . . . , xn) + p\beta f(x1, . . . , xn)
2
\Bigr\}
q +
+
\Bigl\{
p\alpha + 1
\Bigr\}
f(x1, . . . , xn)
2b\prime - pf(x1, . . . , xn)pf(x1, . . . , xn) -
- f(x1, . . . , xn)p
\prime f(x1, . . . , xn). (3)
This implies that \{ b\prime , q, 1\} is linearly C dependent. Then there exist \beta 1, \beta 2, \beta 3 \in C such that
\beta 1b
\prime + \beta 2q + \beta 31 = 0. Again using similar argument as we have used above, since q /\in C, we get
\beta 1 \not = 0 and, hence, b\prime = \alpha \prime q + \beta \prime , where \alpha \prime = - \beta - 1
1 \beta 2 and \beta \prime = - \beta - 1
1 \beta 3. Thus equation (3)
reduces to
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
996 S. K. TIWARI, B. PRAJAPATI
(a\prime + p\gamma + p\alpha \beta \prime + \beta \prime )f(x1, . . . , xn)
2 +
+
\Bigl\{
(a - p+ p\beta + p\alpha \alpha \prime + \alpha \prime )f(x1, . . . , xn)
2 - f(x1, . . . , xn)qf(x1, . . . , xn)
\Bigr\}
q -
- pf(x1, . . . , xn)pf(x1, . . . , xn) - f(x1, . . . , xn)p
\prime f(x1, . . . , xn).
Since \{ q, 1\} is linearly C -independent, hence, U satisfies\Bigl\{
(a - p+ p\beta + p\alpha \alpha \prime + \alpha \prime )f(x1, . . . , xn)
2 - f(x1, . . . , xn)qf(x1, . . . , xn)
\Bigr\}
q = 0,
that is, U satisfies\Bigl\{
(a - p+ p\beta + p\alpha \alpha \prime + \alpha \prime )f(x1, . . . , xn) - f(x1, . . . , xn)q
\Bigr\}
f(x1, . . . , xn)q = 0.
Since \{ q, 1\} is linearly C -independent, hence, U satisfies
f(x1, . . . , xn)qf(x1, . . . , xn)q = 0.
This gives that q \in C, a contradiction.
Next, suppose that \mathrm{h}(x1, . . . , xn) is a non trivial GPI for U. In case C is infinite, we have
\mathrm{h}(x1, . . . , xn) = 0 for all x1, . . . , xn \in U \otimes C C, where C is the algebraic closure of C. Since
both U and U \otimes C C are prime and centrally closed [11] (Theorems 2.5 and 3.5), we may replace
R by U or U \otimes C C according to C finite or infinite. Then R is centrally closed over C and
\mathrm{h}(x1, . . . , xn) = 0 for all x1, . . . , xn \in R. By Martindale’s theorem [19], R is then a primitive ring
with nonzero socle \mathrm{s}\mathrm{o}\mathrm{c}(R) and with C as its associated division ring. Then, by Jacobson’s theorem
[14, p. 75], R is isomorphic to a dense ring of linear transformations of a vector space V over C.
Assume first that V is finite dimensional over C, that is, \mathrm{d}\mathrm{i}\mathrm{m}C V = m. By density of R, we
have R \sim = Mm(C). Since f(r1, . . . , rn) is not central valued on R, R must be non commutative
and so m \geq 2. In this case, by Proposition 3.2, we get that p \in C, a contradiction.
Next we suppose that V is infinite dimensional over C. By Martindale’s theorem [19] (Theorem
3), for any e2 = e \in \mathrm{s}\mathrm{o}\mathrm{c}(R) we have eRe \sim = Mt(C) with t = \mathrm{d}\mathrm{i}\mathrm{m}C V e. Since p and q are not
central, there exist h1, h2 \in \mathrm{S}\mathrm{o}\mathrm{c}(R) such that [p, h1] \not = 0 and [q, h2] \not = 0. By Litoff’s theorem [12],
there exists an idempotent e \in \mathrm{s}\mathrm{o}\mathrm{c}(R) such that ph1, h1p, qh2, h2q, h1, h2 \in eRe. Since R satisfies
generalized identity
e\{ a\prime f(ex1e, . . . , exne)2 + af(ex1e, . . . , exne)
2q + pf(ex1e, . . . , exne)
2b +
+ f(ex1e, . . . , exne)
2b\prime - pf(ex1e, . . . , exne)pf(ex1e, . . . , exne) -
- pf(ex1e, . . . , exne)
2q - f(ex1e, . . . , exne)p
\prime f(ex1e, . . . , exne) -
- f(ex1e, . . . , exne)qf(ex1e, . . . , exne)q\} e,
the subring eRe satisfies
ea\prime ef(x1, . . . , xn)
2 + eaef(x1, . . . , xn)
2eqe+ epef(x1, . . . , xn)
2ebe +
+ f(x1, . . . , xn)
2eb\prime e - epef(x1, . . . , xn)epef(x1, . . . , xn) -
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- epef(x1, . . . , xn)
2eqe - f(x1, . . . , xn)ep
\prime ef(x1, . . . , xn) -
- f(x1, . . . , xn)eqef(x1, . . . , xn)eqe. (4)
Then by the above finite dimensional case, epe and eqe are central elements of eRe. Thus, ph1 =
= (epe)h1 = h1epe = h1p and qh2 = (eqe)h2 = h2(eqe) = h2q, a contradiction.
Lemma 3.1 is proved.
Now we prove Proposition 3.1.
Proof of Proposition 3.1. By the hypothesis, we have
a
\Bigl(
pf(r1, . . . , rn)
2 + f(r1, . . . , rn)
2q
\Bigr)
+
\Bigl(
pf(r1, . . . , rn)
2 + f(r1, . . . , rn)
2q
\Bigr)
b =
=
\Bigl(
pf(r1, . . . , rn) + f(r1, . . . , rn)q
\Bigr) 2
,
that is,
apf(r1, . . . , rn)
2 + af(r1, . . . , rn)
2q + pf(r1, . . . , rn)
2b+ f(r1, . . . , rn)
2qb =
= pf(r1, . . . , rn)pf(r1, . . . , rn) + f(r1, . . . , rn)qf(r1, . . . , rn)q +
+ pf(r1, . . . , rn)
2q + f(r1, . . . , rn)qpf(r1, . . . , rn).
By Lemma 3.1 we have that p \in C and q \in C. Then H(x) = (p+ q)x = \lambda x, where \lambda = p+ q \in C.
From the given hypothesis we get \lambda \{ (a - \lambda )f(r)2 + f(r)2b\} = 0. If \lambda = 0, then H(x) = \lambda x = 0
for all x \in R, which is the conclusion 1. Let \lambda \not = 0. Then we obtain (a - \lambda )f(r)2 + f(r)2b = 0.
From Fact 2.5 we have one of the following:
b \in C and a - \lambda + b = 0, which gives a \in C. Therefore, G(x) = (a+ b)x = \lambda x = H(x) for
all x \in R, which is the conclusion 2.
a - \lambda \in C and a - \lambda + b = 0 which gives b \in C, a \in C. Therefore, G(x) = (a + b)x =
= \lambda x = H(x) for all x \in R, which is the conclusion 2.
a+ b = \lambda and f(x1, . . . , xn)
2 is central valued on R which gives b = \lambda - a.
In this case, we get G(x) = ax+xb = ax+\lambda x - xa = [a, x]+\lambda x for all x \in R and f(x1, . . . , xn)
2
is central valued on R, which is the conclusion 3.
4. Proof of Theorem 1.1. If H = 0, then we are done. Suppose that H \not = 0. In view of [16]
(Theorem 3), we may assume that, for some a, p \in U, there exist derivations d and \delta on U such that
G(x) = ax+ d(x) and H(x) = px+ \delta (x) for all x \in R. Then by the hypothesis, we have
a
\Bigl(
pf(x1, . . . , xn)
2 + \delta (f(x1, . . . , xn)
2)
\Bigr)
+ d
\Bigl(
pf(x1, . . . , xn)
2 +
+ \delta (f(x1, . . . , xn)
2)
\Bigr)
=
\Bigl(
pf(x1, . . . , xn) + \delta (f(x1, . . . , xn))
\Bigr) 2
.
By simplifying above relation, we obtain
apf(x1, . . . , xn)
2 + a\delta (f(x1, . . . , xn)
2) + d(p)f(x1, . . . , xn)
2 +
+ pd(f(x1, . . . , xn))f(x1, . . . , xn) + pf(x1, . . . , xn)d(f(x1, . . . , xn)) +
+ d(\delta (f(x1, . . . , xn)
2)) = pf(x1, . . . , xn)pf(x1, . . . , xn) +
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+ \delta (f(x1, . . . , xn))pf(x1, . . . , xn) +
+ pf(x1, . . . , xn)\delta (f(x1, . . . , xn)) + (\delta (f(x1, . . . , xn)))
2, (5)
that is,
apf(x1, . . . , xn)
2 + a\delta (f(x1, . . . , xn)
2) + d(p)f(x1, . . . , xn)
2 +
+ pd(f(x1, . . . , xn))f(x1, . . . , xn) + pf(x1, . . . , xn)d(f(x1, . . . , xn)) +
+ (d\delta (f(x1, . . . , xn)))f(x1, . . . , xn) + \delta (f(x1, . . . , xn))d(f(x1, . . . , xn)) +
+ d(f(x1, . . . , xn))\delta (f(x1, . . . , xn)) + f(x1, . . . , xn)(d\delta (f(x1, . . . , xn))) =
= pf(x1, . . . , xn)pf(x1, . . . , xn) + \delta (f(x1, . . . , xn))pf(x1, . . . , xn) +
+ pf(x1, . . . , xn)\delta (f(x1, . . . , xn)) + (\delta (f(x1, . . . , xn)))
2. (6)
If d and \delta both are inner derivations then the result follows from Proposition 3.1. So assume that
both d and \delta are not an inner derivations. Now we have the following cases.
Case I. Let d be inner derivation and \delta be an outer derivation. Then, for some q \in U, d(x) =
= [q, x] for all x \in R. From equation (5), we get
apf(x1, . . . , xn)
2 + a\delta (f(x1, . . . , xn))f(x1, . . . , xn) +
+ af(x1, . . . , xn)\delta (f(x1, . . . , xn)) + [q, p]f(x1, . . . , xn)
2 +
+ p
\bigl[
q, f(x1, . . . , xn)
\bigr]
f(x1, . . . , xn) + pf(x1, . . . , xn)
\bigl[
q, f(x1, . . . , xn)
\bigr]
+
+
\bigl[
q, \delta (f(x1, . . . , xn))f(x1, . . . , xn)
\bigr]
+
\bigl[
q, f(x1, . . . , xn)\delta (f(x1, . . . , xn))
\bigr]
=
= pf(x1, . . . , xn)pf(x1, . . . , xn) + pf(x1, . . . , xn)\delta (f(x1, . . . , xn)) +
+ \delta (f(x1, . . . , xn))pf(x1, . . . , xn) + (\delta f(x1, . . . , xn))
2. (7)
In (7) replace \delta (f(x1, . . . , xn)) with f \delta (x1, . . . , xn) +
\sum
i
f(x1, . . . , \delta (xi), . . . , xn):
apf(x1, . . . , xn)
2 + af \delta (x1, . . . , xn)f(x1, . . . , xn) +
+ a
\sum
i
f(x1, . . . , \delta (xi), . . . , xn)f(x1, . . . , xn) + af(x1, . . . , xn)f
\delta (x1, . . . , xn) +
+ af(x1, . . . , xn)
\sum
i
f(x1, . . . , \delta (xi), . . . , xn) + [q, p]f(x1, . . . , xn)
2 +
+ p
\bigl[
q, f(x1, . . . , xn)
\bigr]
f(x1, . . . , xn) + pf(x1, . . . , xn)
\bigl[
q, f(x1, . . . , xn)
\bigr]
+
+
\Biggl[
q, f \delta (x1, . . . , xn)f(x1, . . . , xn) +
\sum
i
f(x1, . . . , \delta (xi), . . . , xn)f(x1, . . . , xn)
\Biggr]
+
+
\Biggl[
q, f(x1, . . . , xn)f
\delta (x1, . . . , xn) + f(x1, . . . , xn)
\sum
i
f(x1, . . . , \delta (xi), . . . , xn)
\Biggr]
=
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GENERALIZED DERIVATIONS ACTING ON MULTILINEAR POLYNOMIALS . . . 999
= pf(x1, . . . , xn)pf(x1, . . . , xn) + pf(x1, . . . , xn)f
\delta (x1, . . . , xn) +
+ pf(x1, . . . , xn)
\sum
i
f(x1, . . . , \delta (xi), . . . , xn) + f \delta (x1, . . . , xn)pf(x1, . . . , xn) +
+
\sum
i
f(x1, . . . , \delta (xi), . . . , xn)pf(x1, . . . , xn) +
+
\Biggl(
f \delta (x1, . . . , xn) +
\sum
i
f(x1, . . . , \delta (xi), . . . , xn)
\Biggr) 2
.
Since \delta is outer, by Kharchenko’s theorem (see Fact 2.4), we replace \delta (xi) by yi in above
expression, we get
apf(x1, . . . , xn)
2 + af \delta (x1, . . . , xn)f(x1, . . . , xn) +
+ a
\sum
i
f(x1, . . . , yi, . . . , xn)f(x1, . . . , xn) + af(x1, . . . , xn)f
\delta (x1, . . . , xn) +
+ af(x1, . . . , xn)
\sum
i
f(x1, . . . , yi, . . . , xn) + [q, p]f(x1, . . . , xn)
2 +
+ p
\bigl[
q, f(x1, . . . , xn)
\bigr]
f(x1, . . . , xn) + pf(x1, . . . , xn)
\bigl[
q, f(x1, . . . , xn)
\bigr]
+
+
\Biggl[
q, f \delta (x1, . . . , xn)f(x1, . . . , xn) +
\sum
i
f(x1, . . . , yi, . . . , xn)f(x1, . . . , xn)
\Biggr]
+
+
\Biggl[
q, f(x1, . . . , xn)f
\delta (x1, . . . , xn) + f(x1, . . . , xn)
\sum
i
f(x1, . . . , yi, . . . , xn)
\Biggr]
=
= pf(x1, . . . , xn)pf(x1, . . . , xn) + pf(x1, . . . , xn)f
\delta (x1, . . . , xn) +
+ pf(x1, . . . , xn)
\sum
i
f(x1, . . . , yi, . . . , xn) + f \delta (x1, . . . , xn)pf(x1, . . . , xn) +
+
\sum
i
f(x1, . . . , yi, . . . , xn)pf(x1, . . . , xn) +
+
\Biggl(
f \delta (x1, . . . , xn) +
\sum
i
f(x1, . . . , yi, . . . , xn)
\Biggr) 2
(8)
for all xi, yi \in U. In particular, for x1 = 0 in relation (8), we obtain f(y1, x2, . . . , xn)
2 = 0, a
contradiction.
Case II. Let d be an outer derivation on R and \delta be an inner derivation on R. For some q \in U
such that \delta (x) = [q, x] for all x \in R. Then (5) implies that
apf(x1, . . . , xn)
2 + a
\bigl[
q, f(x1, . . . , xn)
2
\bigr]
+ d(p)f(x1, . . . , xn)
2 +
+ pd(f(x1, . . . , xn))f(x1, . . . , xn) + pf(x1, . . . , xn)d(f(x1, . . . , xn)) +
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1000 S. K. TIWARI, B. PRAJAPATI
+
\bigl[
d(q), f(x1, . . . , xn)
2
\bigr]
+
\bigl[
q, d(f(x1, . . . , xn)
2)
\bigr]
=
= pf(x1, . . . , xn)pf(x1, . . . , xn) +
\bigl[
q, f(x1, . . . , xn)
\bigr]
pf(x1, . . . , xn) +
+ pf(x1, . . . , xn)
\bigl[
q, f(x1, . . . , xn)
\bigr]
+
\Bigl( \bigl[
q, f(x1, . . . , xn)
\bigr] \Bigr) 2
. (9)
This can be rewritten as
apf(x1, . . . , xn)
2 + a
\bigl[
q, f(x1, . . . , xn)
2
\bigr]
+ d(p)f(x1, . . . , xn)
2 +
+ pd(f(x1, . . . , xn))f(x1, . . . , xn) + pf(x1, . . . , xn)d(f(x1, . . . , xn)) +
+
\bigl[
d(q), f(x1, . . . , xn)
2
\bigr]
+
\bigl[
q, d(f(x1, . . . , xn))f(x1, . . . , xn)
\bigr]
+
+
\bigl[
q, f(x1, . . . , xn)d(f(x1, . . . , xn))
\bigr]
=
= pf(x1, . . . , xn)pf(x1, . . . , xn) +
\bigl[
q, f(x1, . . . , xn)
\bigr]
pf(x1, . . . , xn) +
+pf(x1, . . . , xn)
\bigl[
q, f(x1, . . . , xn)
\bigr]
+
\Bigl( \bigl[
q, f(x1, . . . , xn)
\bigr] \Bigr) 2
. (10)
Since d is an outer derivation on R, in (10) replace d(f(x1, . . . , xn)) with fd(x1, . . . , xn) +
+
\sum
i
f(x1, . . . , yi, . . . , xn), where d(xi) = yi, we obtain
apf(x1, . . . , xn)
2 + a
\bigl[
q, f(x1, . . . , xn)
2
\bigr]
+ d(p)f(x1, . . . , xn)
2 +
+ pfd(x1, . . . , xn)f(x1, . . . , xn) + p
\sum
i
f(x1, . . . , yi, . . . , xn)f(x1, . . . , xn) +
+ pf(x1, . . . , xn)f
d(x1, . . . , xn) + pf(x1, . . . , xn)
\sum
i
f(x1, . . . , yi, . . . , xn) +
+
\Bigl[
d(q), f(x1, . . . , xn)
2
\Bigr]
+
\Biggl[
q, fd(x1, . . . , xn)f(x1, . . . , xn) +
+
\sum
i
f(x1, . . . , yi, . . . , xn)f(x1, . . . , xn)
\Biggr]
+
+
\Biggl[
q, f(x1, . . . , xn)f
d(x1, . . . , xn) + f(x1, . . . , xn)
\sum
i
f(x1, . . . , yi, . . . , xn)
\Biggr]
=
= pf(x1, . . . , xn)pf(x1, . . . , xn) +
\bigl[
q, f(x1, . . . , xn)
\bigr]
pf(x1, . . . , xn) +
+ pf(x1, . . . , xn)
\bigl[
q, f(x1, . . . , xn)
\bigr]
+
\Bigl( \bigl[
q, f(x1, . . . , xn)
\bigr] \Bigr) 2
.
Hence, U satisfies the blended component
p
\sum
i
f(x1, . . . , yi, . . . , xn)f(x1, . . . , xn) + pf(x1, . . . , xn)
\sum
i
f(x1, . . . , yi, . . . , xn) +
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GENERALIZED DERIVATIONS ACTING ON MULTILINEAR POLYNOMIALS . . . 1001
+
\Biggl[
q,
\sum
i
f(x1, . . . , yi, . . . , xn)f(x1, . . . , xn)
\Biggr]
+
+
\Biggl[
q, f(x1, . . . , xn)
\sum
i
f(x1, . . . , yi, . . . , xn)
\Biggr]
= 0.
In particular, for y1 = x1 and y2 = . . . = yn = 0, we obtain 2pf(x1, . . . , xn)
2 + 2
\bigl[
q, f(x1, . . .
. . . , xn)
2
\bigr]
= 0. Since char(R) \not = 2, it implies that pf(x1, . . . , xn)2 +
\bigl[
q, f(x1, . . . , xn)
2
\bigr]
= 0. This
gives that
(p+ q)f(x1, . . . , xn)
2 - f(x1, . . . , xn)
2q = 0.
By Fact 2.5, we have one of the following:
q \in C and p = 0, which implies that H = 0, a contradiction.
p+ q \in C and p = 0, which gives that q \in C. In this case H = 0, a contradiction.
p = 0 and f(x1, . . . , xn)
2 is a central valued on R. By using the fact that if z \in Z(R), then
d(z) \in Z(R), where d is a derivation on R, the equation (9) implies that [q, f(x1, . . . , xn)]2 = 0.
By [9] (Theorem 1.1), we get q \in C which implies that H = 0, a contradiction.
Case III. Let none of d and \delta be inner derivations on R. We have the following two subcases.
Subcase I. Suppose that d and \delta are C -dependent modulo inner derivation of U, that is, \alpha d +
+ \beta \delta = adq, where \alpha , \beta \in C, q \in U and adq(x) = [q, x] for all x \in U. If \alpha = 0, then \delta is
inner derivation on R, a contradiction. If \beta = 0, then d is inner derivation on R, a contradiction.
Hence, \alpha and \beta both can not be zero. This gives that d(x) = \beta 1\delta (x) + [q\prime , x] for all x \in R, where
\beta 1 = - \alpha - 1\beta and q\prime = \alpha - 1q. Thus, from (5), we have
apf(x1, . . . , xn)
2 + a\delta (f(x1, . . . , xn))f(x1, . . . , xn) +
+ af(x1, . . . , xn)\delta (f(x1, . . . , xn)) + d(p)f(x1, . . . , xn)
2 + p
\Bigl(
\beta 1\delta (f(x1, . . . , xn)) +
+
\bigl[
q\prime , f(x1, . . . , xn)
\bigr] \Bigr)
f(x1, . . . , xn) + pf(x1, . . . , xn)
\Bigl(
\beta 1\delta (f(x1, . . . , xn)) +
+
\bigl[
q\prime , f(x1, . . . , xn)
\bigr] \Bigr)
+ \beta 1\delta
2(f(x1, . . . , xn))f(x1, . . . , xn) +
+ \beta 1f(x1, . . . , xn)\delta
2(f(x1, . . . , xn)) + 2\beta 1\delta (f(x1, . . . , xn))
2 +
+
\Bigl[
q\prime , \delta (f(x1, . . . , xn))f(x1, . . . , xn) + f(x1, . . . , xn)\delta (f(x1, . . . , xn))
\Bigr]
=
= pf(x1, . . . , xn)pf(x1, . . . , xn) + \delta (f(x1, . . . , xn))pf(x1, . . . , xn) +
+ pf(x1, . . . , xn)\delta (f(x1, . . . , xn)) + (\delta (f(x1, . . . , xn)))
2. (11)
First, we can replace \delta (f(x1, . . . , xn)) with f \delta (x1, . . . , xn) +
\sum
i
f(x1, . . . , yi, . . . , xn) and
\delta 2(f(x1, . . . , xn)) with
f \delta 2(x1, . . . , xn) + 2
\sum
i
f \delta (x1, . . . , yi, . . . , xn) +
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1002 S. K. TIWARI, B. PRAJAPATI
+
\sum
i
f(x1, . . . , wi, . . . , xn) +
\sum
i \not =j
f(x1, . . . , yi, . . . , yj , . . . , xn),
where \delta (xi) = yi and \delta 2(xi) = wi in (11) and then U satisfies the blended component
\beta 1
\Biggl( \sum
i
f(x1, . . . , wi, . . . , xn)
\Biggr)
f(x1 . . . , xn) +
+ \beta 1f(x1, . . . , xn)
\Biggl( \sum
i
f(x1, . . . , wi, . . . , xn)
\Biggr)
= 0 (12)
for all x1, . . . , xn \in R and wi \in R. In particular, for w1 = x1 and w2 = . . . = wn = 0, we obtain
2\beta 1f(x1, . . . , xn)
2 = 0. Since char(R) \not = 2, it implies that \beta 1 = 0. Then d is an inner derivation, a
contradiction.
Subcase II. Suppose that d and \delta are C -independent modulo inner derivation of U. By using
Kharchenko’s theorem (see Fact 2.4), we can replace d(f(x1, . . . , xn)) with fd(x1, . . . , xn) +
+
\sum
i
f(x1, . . . , yi, . . . , xn), \delta (f(x1, . . . , xn)) with f \delta (x1, . . . , xn)+
\sum
i
f(x1, . . . , zi, . . . , xn) and
d\delta (f(x1, . . . , xn)) with fd\delta (x1, . . . , xn)+
\sum
i
fd(x1, . . . , zi, . . . , xn)+
\sum
i
f \delta (x1, . . . , yi, . . . , xn)+
+
\sum
i
f(x1, . . . , wi, . . . , xn) +
\sum
i \not =j
f(x1, . . . , yi, . . . , zj , . . . , xn), where d(xi) = yi, \delta (xi) = zi
and d\delta (xi) = wi in equation (6) and then U satisfies the blended component\Biggl( \sum
i
f(x1, . . . , wi, . . . , xn)
\Biggr)
f(x1, . . . , xn) +
+ f(x1, . . . , xn)
\Biggl( \sum
i
f(x1, . . . , wi, . . . , xn
\Biggr)
= 0. (13)
Equation (13) is similar to equation (12), we get a contradiction.
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ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
GENERALIZED DERIVATIONS ACTING ON MULTILINEAR POLYNOMIALS . . . 1003
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Received 07.05.20,
after revision — 05.01.21
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
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| id | umjimathkievua-article-6108 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:26:05Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0d/e703e4adb8caa66d78ab5e87d656d30d.pdf |
| spelling | umjimathkievua-article-61082022-10-24T09:23:12Z Generalized derivations acting on multilinear polynomials as a Jordan homomorphisms Generalized derivations acting on multilinear polynomials as a Jordan homomorphisms Tiwari, S. K. Prajapati, B. Tiwari, S. K. Prajapati, B. Jordan homomorphism, generalized derivations, multilinear polynomials, extended centroid, Utumi quotient ring. UDC 512.5 Let $R$ be a prime ring whose characteristic is not equal to $2,$ let  $U$ be the Utumi quotient ring of $R,$ and let $C$ be the extended centroid of $R.$  Also let $G$ and $H$ be two generalized derivations on $R$ and let $f(x_1,\ldots,x_n)$ be a noncentral multilinear polynomial over $C.$  If $G(H(u^2))=(H(u))^2$ for all $u=f(r_1,\ldots,r_n),$ $r_1,\ldots,r_n \in R,$ then one of the following holds: 1) $H=0;$ 2) there exists $\lambda\in C$ such that $G(x)=H(x)=\lambda x$ for all $x\in R;$ 3) there exist $\lambda\in C$ and $a\in U$ such that $H(x)=\lambda x$ and $G(x)=[a, x]+\lambda x$ for all $x\in R$ and $f(x_1,\ldots,x_n)^2$ is central-valued on $R.$ УДК 512.5Узагальненi похiднi, що дiють на мультилiнiйних полiномахяк жордановi гомоморфiзми Нехай $R$ — просте кільце з характеристикою, що не дорівнює $2,$ $U$ —  фактор-кільце Утумі для $R,$ а $C$ — продовжений центроїд для $R.$  Крім того, припустимо, що $G$ та $H$ — дві узагальнені похідні на $R,$ а $f(x_1,\ldots,x_n)$ —  нецентральний мультилінійний поліном над $C.$  Якщо $G(H(u^2))=(H(u))^2$ для всіх $u=f(r_1,\ldots,r_n),$ $r_1,\ldots,r_n \in R,$ то справджується одне з таких тверджень: 1) $H=0;$ 2) існує таке $\lambda\in C,$ що $G(x)=H(x)=\lambda x$ для всіх $x\in R;$ 3) існують  такі $\lambda\in C$ та $a\in U,$ що $H(x)=\lambda x,$ $G(x)=[a, x]+\lambda x$ для всіх $x\in R$ і $f(x_1,\ldots,x_n)^2$ є центральнозначним на $R.$ Institute of Mathematics, NAS of Ukraine 2022-08-09 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6108 10.37863/umzh.v74i7.6108 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 7 (2022); 991 - 1003 Український математичний журнал; Том 74 № 7 (2022); 991 - 1003 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6108/9281 Copyright (c) 2022 S. K. Tiwari, B Prajapati |
| spellingShingle | Tiwari, S. K. Prajapati, B. Tiwari, S. K. Prajapati, B. Generalized derivations acting on multilinear polynomials as a Jordan homomorphisms |
| title | Generalized derivations acting on multilinear polynomials as a Jordan homomorphisms |
| title_alt | Generalized derivations acting on multilinear polynomials as a Jordan homomorphisms |
| title_full | Generalized derivations acting on multilinear polynomials as a Jordan homomorphisms |
| title_fullStr | Generalized derivations acting on multilinear polynomials as a Jordan homomorphisms |
| title_full_unstemmed | Generalized derivations acting on multilinear polynomials as a Jordan homomorphisms |
| title_short | Generalized derivations acting on multilinear polynomials as a Jordan homomorphisms |
| title_sort | generalized derivations acting on multilinear polynomials as a jordan homomorphisms |
| topic_facet | Jordan homomorphism generalized derivations multilinear polynomials extended centroid Utumi quotient ring. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6108 |
| work_keys_str_mv | AT tiwarisk generalizedderivationsactingonmultilinearpolynomialsasajordanhomomorphisms AT prajapatib generalizedderivationsactingonmultilinearpolynomialsasajordanhomomorphisms AT tiwarisk generalizedderivationsactingonmultilinearpolynomialsasajordanhomomorphisms AT prajapatib generalizedderivationsactingonmultilinearpolynomialsasajordanhomomorphisms |