Lie and Jordan structures of differentially semiprime rings
Properties of Lie and Jordan rings (denoted respectively by RL and RJ) associated with an associative ring R are discussed. Results on connections between the differentially simplicity (respectively primeness, semiprimeness) of R, RL and RJ are obtained.
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| Цитувати: | Lie and Jordan structures of differentially semiprime rings / O.D. Artemovych, M.P. Lukashenko // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 1. — С. 13-31 . — Бібліогр.: 23 назв. — англ. |
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Artemovych, O.D. Lukashenko, M.P. 2019-06-15T20:01:23Z 2019-06-15T20:01:23Z 2015 Lie and Jordan structures of differentially semiprime rings / O.D. Artemovych, M.P. Lukashenko // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 1. — С. 13-31 . — Бібліогр.: 23 назв. — англ. 1726-3255 2010 MSC:Primary 16W25, 16N60; Secondary 17B60, 17C50. https://nasplib.isofts.kiev.ua/handle/123456789/154758 Properties of Lie and Jordan rings (denoted respectively by RL and RJ) associated with an associative ring R are discussed. Results on connections between the differentially simplicity (respectively primeness, semiprimeness) of R, RL and RJ are obtained. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Lie and Jordan structures of differentially semiprime rings Article published earlier |
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Lie and Jordan structures of differentially semiprime rings Artemovych, O.D. Lukashenko, M.P. |
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Properties of Lie and Jordan rings (denoted respectively by RL and RJ) associated with an associative ring R are discussed. Results on connections between the differentially simplicity (respectively primeness, semiprimeness) of R, RL and RJ are obtained.
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1726-3255 |
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Lie and Jordan structures of differentially semiprime rings / O.D. Artemovych, M.P. Lukashenko // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 1. — С. 13-31 . — Бібліогр.: 23 назв. — англ. |
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AT artemovychod lieandjordanstructuresofdifferentiallysemiprimerings AT lukashenkomp lieandjordanstructuresofdifferentiallysemiprimerings |
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2025-11-25T22:29:15Z |
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2025-11-25T22:29:15Z |
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1850563564855623680 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 20 (2015). Number 1, pp. 13–31
© Journal “Algebra and Discrete Mathematics”
Lie and Jordan structures of differentially
semiprime rings
Orest D. Artemovych and Maria P. Lukashenko
Communicated by A. P. Petravchuk
Abstract. Properties of Lie and Jordan rings (denoted
respectively by RL and RJ) associated with an associative ring
R are discussed. Results on connections between the differentially
simplicity (respectively primeness, semiprimeness) of R, RL and RJ
are obtained.
1. Introduction
Throughout here, R is an associative ring (with respect to the addition
“+” and the multiplication “ · ”) with an identity, Der R is the set of all
derivations in R. On the set R we consider two operations: the Lie
multiplication “[−, −]” and the Jordan multiplication “(−, −)” defined
by the rules
[a, b] = a · b − b · a
and
(a, b) = a · b + b · a
for any a, b ∈ R. Then
RL = (R, +, [−, −])
is a Lie ring and
RJ = (R, +, (−, −))
2010 MSC: Primary 16W25, 16N60; Secondary 17B60, 17C50.
Key words and phrases: Derivation, semiprime ring, Lie ring.
14 Lie and Jordan structures
is a Jordan ring (see [13] and [14]) associated with the associative ring R.
Recall that an additive subgroup A of R is called:
• a Lie ideal of R if
[a, r] ∈ A,
• a Jordan ideal of R if
(a, r) ∈ A
for all a ∈ A and r ∈ R. Obviously, A is a Lie (respectively Jordan) ideal
of R if and only if AL (respectively AJ) is an ideal of RL (respectively
RJ).
In all that follows ∆ will be any subset of Der R (in particular, ∆ = {0})
and δ ∈ Der R. A subset K of R is called ∆-stable if d(a) ∈ K for all
d ∈ ∆ and a ∈ K. An ideal I of a (Lie, Jordan or associative) ring A is
said to be a ∆-ideal if I is ∆-stable. A (Lie, Jordan or associative) ring
A is said to be:
• simple (respectively ∆-simple) if there no two-sided ideals (respec-
tively ∆-ideals) other 0 or A,
• prime (respectively ∆-prime) if, for all two-sided ideals (respectively
∆-ideals) K, S of A, the condition KS = 0 implies that K = 0 or
S = 0 (if ∆ = {δ} and A is ∆-prime, then we say that A is δ-prime),
• semiprime (respectively ∆-semiprime) if, for any two-sided ideal
(respectively ∆-ideal) K of A, the condition K2 = 0 implies that
K = 0,
• primary if, for any two-sided ideals K, S of A, the condition KS = 0
implies that K = 0 or S is nilpotent.
Every non-commutative ∆-simple ring is ∆-prime and every ∆-prime
ring is ∆-semiprime. We say that R is Z-torsion-free if, for any r ∈ R
and integers n, the condition nr = 0 holds if and only if r = 0. If the
implication
2r = 0 ⇒ r = 0
is true for any r ∈ R, then R is said to be 2-torsion-free. Let
Fp(R) = {a ∈ R | a has an additive order pk
for some non-negativek = k(a)}
be the p-part of R, where p is a prime. Then Fp(R) is a ∆-ideal of R. If
R is ∆-semiprime, then
pFp(R) = 0.
O. D. Artemovych, M. P. Lukashenko 15
In particular, in a ∆-prime ring R it holds Fp(R) = 0 (and so the
characteristic char R = 0) or Fp(R) = R (and therefore char R = p).
Obviously that the additive group R+ of a ∆-prime ring R is torsion-free
if and only if char R = 0. Recall that a ring R is said to be of bounded
index m, if m is the least positive integer such that xm = 0 for all nilpotent
elements x ∈ R. We say that a ring R satisfies the condition (X) if one of
the following holds:
(1) R or R/P(R) is Z-torsion-free, where P(R) is the prime radical of
R,
(2) R is of bounded index m such that an additive order of every nonzero
torsion element of R, if any, is strictly larger than m.
As noted in [16, p.283], a Z-torsion-free δ-prime ring is semiprime. In
this way we prove the following
Proposition 1. For a ring R the following hold:
(1) if R is a ∆-semiprime ring with the condition (X), then it is
semiprime,
(2) if R is both semiprime (respectively satisfies the condition (X)) and
∆-prime, then R is prime.
Relations between properties of an associative ring R, a Lie ring RL
and a Jordan ring RJ was studied by I.N. Herstein and his students (see
[7, 8, 11] and bibliography in [9] and [5]); he has obtained, for a ring R
of characteristic different from 2, that the simplicity of R implies the
simplicity of a Jordan ring RJ [7, Theorem 1], and also that every Lie
ideal of a simple Lie ring R is contained in the center Z(R) [7, Theorem
3]. K. McCrimmon [20, Theorem 4] has proved that R is a simple algebra
if and only if RJ is a simple Jordan algebra. Our result is the following
Theorem 1. For a 2-torsion-free ring R the following statements are
true:
(1) R is a ∆-simple ring if and only if RJ is a ∆-simple Jordan ring,
(2) R is a ∆-prime ring if and only if RJ is a ∆-prime Jordan ring,
(3) R is a ∆-semiprime ring if and only if RJ is a ∆-semiprime Jordan
ring.
16 Lie and Jordan structures
Let us d ∈ ∆. Since C(R) and ann C(R) are ∆-ideals, the rule
d : R/ ann C(R) ∋ r + ann C(R) 7→ d(r) + ann C(R) ∈ R/ ann C(R)
determines a derivation d of the quotient ring R/ ann C(R). Then
∆ = {d | d ∈ ∆} ⊆ Der(R/ ann C(R)).
Inasmuch d(Z(R)) ⊆ Z(R), the rule
d̂ : RL/Z(R) ∋ r + Z(R) 7→ d(r) + Z(R) ∈ RL/Z(R)
determines a derivation d̂ of the Lie ring RL/Z(R). Then
∆̂ = {d̂ | d ∈ ∆} ⊆ Der(RL/Z(R)).
Since the center Z(R) is a nonzero Lie ideal of an associative ring R with
an identity, a Lie ring RL is not ∆-simple. Our next result is the following
Theorem 2. Let R be a 2-torsion-free ring. Then the following are true:
(1) if the quotient ring RL/Z(R) is a ∆̂-simple Lie ring, then R is
non-commutative and R/ ann C(R) is a ∆-simple ring,
(2) if R is a ∆-simple ring, then RL/Z(R) is a ∆̂-simple Lie ring or
R is commutative,
(3) if RL/Z(R) is a ∆̂-semiprime Lie ring, then R is non-commutative
and the quotient ring R/ ann C(R) is a ∆-semiprime ring,
(4) if R is a ∆-semiprime ring, then RL/Z(R) is a ∆̂-semiprime Lie
ring or R is commutative,
(5) if RL/Z(R) is a ∆̂-prime Lie ring, then R is non-commutative and
R/ ann C(R) is a ∆-prime ring,
(6) if R is a ∆-prime ring, then RL/Z(R) is a ∆̂-prime Lie ring or R
is commutative.
Throughout, let Z(R) denote the center of R, [A, B] (respectively
(A, B)) an additive subgroup of R generated by all commutators [a, b]
(respectively (a, b)), where a ∈ A and b ∈ B, C(R) the commutator ideal
of R, N(R) the set of nilpotent elements in R, char R the characteristic of
R, annl I = {a ∈ R | aI = 0} the left annihilator of I in R, annr I = {a ∈
R | Ia = 0} the right annihilator of I in R, ann I = (annr I) ∩ (annl I),
CR(I) = {a ∈ R | ai = ia for all i ∈ I} the centralizer of I in R and
∂a(x) = [a, x] for a, x ∈ R.
All other definitions and facts are standard and it can be found in
[10], [17] and [19].
O. D. Artemovych, M. P. Lukashenko 17
2. Differentially prime right Goldie rings
Let agree that
d0 = idR
is the identity endomorphism for d ∈ ∆.
Lemma 1. The following conditions are equivalent:
(1) R is a ∆-semiprime ring,
(2) for any ∆-ideals A, B of R the implication
AB = 0 ⇒ A ∩ B = 0
is true,
(3) if a ∈ R is such that
aRδm1
1 . . . δmk
k (a) = 0
for any integers k > 1, mi > 0 and derivations δi ∈ ∆ (i = 1, . . . , k),
then a = 0.
Proof. A simple modification of Proposition 2 from [17, §3.2].
Lemma 2. The following conditions are equivalent:
(1) R is a ∆-prime ring,
(2) a left annihilator annl I of a left ∆-ideal I of R is zero,
(3) a right annihilator annr I of a right ∆-ideal I of R is zero,
(4) if a, b ∈ R are such that
aRδm1
1 . . . δmk
k (b) = 0
for any integers k > 1, mj > 0 and derivations δj ∈ ∆ (j = 1, . . . , k),
then a = 0 or b = 0.
Proof. A simple consequence of Lemma 2.1.1 from [10].
If I is an ideal of a ring R, then
CR(I) = {x ∈ R | x + I is regular in the quotient ring R/I}
(see [19, Chapter 2, §1]). The next lemma extends Proposition 1 of [15].
18 Lie and Jordan structures
Lemma 3. Let R be a right Goldie ring and δ ∈ Der R. If R is δ-prime,
then:
(a) the set N = N(R) of nilpotent elements of R is its prime radical,
(b)
⋂k
i=1 δ−1(N) = 0 for some integer k,
(c) CR(0) = CR(N).
Proof. From Theorem 2.2 of [16] (see the part (ii) ⇒ (iii) of its proof), we
obtain (a) and (b). By Proposition 4.1.3 of [19], CR(0) ⊆ CR(N). By the
same argument as in [16, p.284], we can obtain that CR(0) = CR(N).
Corollary 1. If R is a commutative δ-prime Goldie ring and δ ∈ Der R,
then N(R) contains all zero-divisors of R.
By Corollary 1.4 of [6], if I is a δ-prime ideal of a right Noetherian
ring R and R/I has characteristic 0, then I is prime. The following lemma
is an extension of Lemma 2.5 from [6].
Lemma 4. Let R be a 2-torsion-free commutative Goldie ring and δ ∈
Der R. If R is δ-prime, then it is an integral domain.
Proof. Assume that a ∈ ann N(R), b ∈ N(R) and r ∈ R. Then
0 = δ2(arb) = δ(δ(a)rb + aδ(r)b + arδ(b))
= δ2(a)rb + 2δ(a)δ(r)b + 2δ(a)rδ(b) + aδ2(r)b + 2aδ(r)δ(b) + arδ2(b)
and so
2δ(a)Rδ(b) ⊆ N(R).
This means that δ(a) ∈ N(R) or δ(b) ∈ N(R). Hence N(R) is δ-stable.
By Lemma 3, N(R) is a ideal and therefore N(R) = 0. By Lemma 1.2 of
[4], R is prime and consequently it is an integral domain.
Proof of Proposition 1.
(1) By Proposition 1.3 of [6] and Theorem 1 of [1], the prime radical
P(R) is a ∆-ideal and so P(R) = 0 is zero.
(2) Since P(R) = 0, R is prime by Lemma 1.2 from [4].
By Theorem 4 of [22], a ∆-simple ring R of characteristic 0 is prime.
Since every non-commutative ∆-simple ring is ∆-prime, in view of Propo-
sition 1 we obtain the following
Corollary 2. Let R be a semiprime ring (respectively a ring R satisfy
the condition (X)). If R is ∆-simple, then it is prime.
O. D. Artemovych, M. P. Lukashenko 19
3. Differential analogues of Herstein’s results
For the proof of Theorem 2 we need the next results. In the proofs
below we use the same consideration, as in [12, Chapter 1, §1], and present
them here in order to have the paper more self-contained. Let agree that
everywhere in this section k > 1 and mi > 0 are integers (i = 1, . . . , k).
Lemma 5. Let R be a ∆-semiprime ring, A and B its ∆-ideals. Then
the following statements hold:
(i) if AB = 0, then BA = 0.
(ii) annl A = annr A.
(iii) A ∩ annr A = 0.
Proof. (i) Indeed, BA is a ∆-ideal and (BA)2 = 0 and so BA = 0.
(ii) We denote (annr A)A by X. Since X is a ∆-ideal and X2 = 0, we
deduce that X = 0. This means that
annr A ⊆ annl A.
The inverse inclusion we can prove similarly.
(iii) Since A ∩ annr A is a nilpotent ∆-ideal, the assertion holds.
Henceforth
Xa = {[δm1
1 . . . δmk
k (a), x] | x ∈ R, δi ∈ ∆, mi > 0
and k > 1 are integers (i = 1, . . . , k)}.
It is clear that [a, x] ∈ Xa.
Lemma 6. Let R be a ∆-semiprime ring and a ∈ R. Then the following
statements hold:
(i) if
a[δm1
1 . . . δmk
k (a), R] = 0
for any integers k > 1, mi > 0 and derivations δi ∈ ∆ (i = 1, . . . , k),
then a ∈ Z(R),
(ii) if I is a right ∆-ideal of R, then Z(I) ⊆ Z(R),
(iii) if I is a commutative right ∆-ideal of R and I is nonzero, then
I ⊆ Z(R). If, moreover, R is ∆-prime, then it is commutative.
20 Lie and Jordan structures
Proof. (i) Let x, y ∈ R and d, δ ∈ ∆. Since
[b, xy] = [b, x]y + x[b, y] (3.1)
for any b ∈ Xa and a[b, xy] = 0, we conclude that ax[b, y] = 0. This gives
that ayx[b, y] = 0 and yax[b, y] = 0 and consequently
(R[a, y]R)2 = 0. (3.2)
In addition,
0 = d(a[b, x]) = d(a)[b, x].
Multiplying (3.1) by d(a) on left we get d(a)x[b, y] = 0. Moreover,
0 = δ(ax[d(b), y]) = δ(a)x[d(b), y]
and, by the similar argument, we obtain that
δm1
1 . . . δmk
k (a)x[δm1
1 . . . δmk
k (a), y] = 0
for any integers k > 1, mi > 0 and derivations δi ∈ ∆ (i = 1, . . . , k). As
in the proof of the condition (3.2), we deduce that
(R[δm1
1 . . . δmk
k (a), y]R)2 = 0.
Then
I =
∞∑
k=1
∑
δ1...δk∈∆
y∈R
R[δm1
1 . . . δmk
k (a), y]R
is a sum of nilpotent ideals and therefore it is a nil ideal. Since I is a
∆-ideal, we conclude that I = 0 and, as a consequence, a ∈ Z(R).
(ii) Let a ∈ Z(I) and y ∈ R. Then, for δ1, . . . , δk ∈ ∆, we have
δm1
1 . . . δmk
k (a) ∈ Z(I)
and ay ∈ I. This gives that
a(δm1
1 . . . δmk
k (a)y) = δm1
1 . . . δmk
k (a)(ay) = a(yδm1
1 . . . δmk
k (a)),
and thus
a[δm1
1 . . . δmk
k (a), y] = 0.
By (i), a ∈ Z(R) is central.
O. D. Artemovych, M. P. Lukashenko 21
(iii) By (ii), I ⊆ Z(R). Assume that R is ∆-prime, u, v ∈ R and
a ∈ I. Then au ∈ I and so au ∈ Z(R). Since
a(uv) = (au)v = v(au) = (va)u = a(vu),
we see that
[u, v] ∈ annr I.
By Lemma 2(3), [u, v] = 0 and hence R is commutative.
Lemma 7. Let R be a ∆-prime ring and a ∈ R. If a ∈ CR(I) for some
nonzero right ∆-ideal I of R, then a ∈ Z(R).
Proof. Let us y ∈ R and b ∈ I. Then by ∈ I and so bay = a(by) = bya.
This yields that
I[a, y] = 0 = [a, y]I.
By Lemma 2(3), [a, y] = 0. Hence a ∈ Z(R).
Lemma 8. The left annihilator annl(Xa) is a left ∆-ideal of R.
Proof. Immediate from the definition.
Lemma 9. If R is a ∆-semiprime ring, then CR([R, R]) ⊆ Z(R).
Proof. Let us a ∈ CR([R, R]), d, δ ∈ ∆ and x, y ∈ R. Putting x for a and
xd(a) for xy in (3.1) we obtain
[x, xd(a)] = [x, x]d(a) + x[x, d(a)]
and, as a consequence, [a, x[x, d(a)]] = 0 and [a, x][x, d(a)] = 0. Then,
by the same reasons as in the proof of Lemma 6(i), we obtain that
[a, x] ∈ annl(Xa) and A = annl(Xa) is a ∆-ideal. Then
[δ(a), x][d(a), x] = δ([a, x][d(a), x]) = 0.
Since A ∩ annl A = 0, we deduce that is a nilpotent ∆-ideal and so
a ∈ Z(R).
Lemma 10. Let R be a 2-torsion-free ∆-semiprime ring. If a ∈ R
commutes with all elements of Xa, then a ∈ Z(R).
22 Lie and Jordan structures
Proof. Let r, x, y ∈ R and d ∈ ∆. It is clear that ∂2
a(x) = 0. From
∂2
a(xy) = 0 it follows that
2∂a(x)∂a(y) = 0
and so ∂a(x)∂a(y) = 0. Since
0 = ∂a(x)∂a(rx) = ∂a(x)∂a(r)x + ∂a(x)r∂a(x) = ∂a(x)r∂a(x),
we deduce that ∂a(x)R∂a(x) = 0 and (∂a(x)R)2 = 0. Moreover, a[b, x] =
[b, x]a for any [b, x] ∈ Xa and therefore
d(a)[b, x] + a[d(b), x] + a[b, d(x)] = [b, x]d(a) + [d(b), x]a + [b, d(x)]a.
From this it holds that
d(a)[b, x] = [b, x]d(a).
This means that CR(Xa) is ∆-stable and (∂d(a)(x)R)2 = 0. As a conse-
quence,
I =
∞∑
k=1
∑
x∈R
mk>0
δ1,...,δk∈∆
∂δ
m1
1
...δ
mk
k
(a)(x)R
is a sum of nilpotent ideals and so I is a nil ideal. Since I is a ∆-ideal,
we deduce that I = 0. Hence a ∈ Z(R).
The next lemma is an extension of Lemma 1 from [11] in the differential
case.
Lemma 11. Let R be a 2-torsion-free ∆-semiprime ring, T its Lie ∆-
ideal. If [T, T ] ⊆ Z(R), then T ⊆ Z(R).
Proof. Let x ∈ R and t ∈ T .
1) If [T, T ] = 0, then [t, x] ∈ T and so [t, [t, x]] = 0. By Lemma 10,
T ⊆ Z(R).
2) Now assume that 0 6= [a, b] ∈ [T, T ] for some a, b ∈ T . Then
∂a(b) ∈ Z(R) and ∂2
a(R) ⊆ Z(R).
Moreover, we have that
Z(R) ∋ ∂2
a(bx) = ∂a(∂a(b)x + b∂a(x))
= ∂2
a(b)x + 2∂a(b)∂a(x) + b∂2
a(x)
= 2∂a(b)∂a(x) + b∂2
a(x)
O. D. Artemovych, M. P. Lukashenko 23
and hence
[2∂a(b)∂a(x) + b∂2
a(x), b] = 0.
Then
0 = 2∂b(∂a(b))∂a(x) + 2∂a(b)∂b(∂a(x)) + ∂b(b)∂2
a(x) + b∂b(∂
2
a(x))
= 2∂a(b)∂b(∂a(x))
(3.3)
and
∂a(ba) = ∂a(b)a + b∂a(a) = ∂a(b)a.
Replacing ba for x in (3.3) we have
0 = 2∂a(b)∂b(∂a(b)a) = 2∂a(b)(∂b(∂a(b)) + ∂a(b)∂b(a)) = −2∂a(b)3
and thus ∂a(b)3 = 0. Then R∂a(b) is a nilpotent ideal in R and, as a
consequence, ∑
a,b∈T
R∂a(b)
is a nonzero nil ∆-ideal, a contradiction.
Lemma 12. If U is a Lie ∆-ideal of a ring R and I(U) = {u ∈ R |
uR ⊆ U}, then I(U) is the largest ∆-ideal of R such that I(U) ⊆ U .
Proof. Let u, v ∈ I(U), x, y ∈ R and δ ∈ ∆. Clearly that I(U) is an
additive subgroup of R, I(U) ⊆ U and (ux)y = u(xy) ∈ (ux)R = u(xR) ⊆
uR ⊆ U that is ux ∈ I(U). From
u(xy) − (yu)x = (ux)y − y(ux) = [ux, y] ∈ U
(and so (yu)x ∈ U) it holds that yu ∈ I(U). Hence U is a two-sided ideal
of R. Moreover,
δ(u)x + uδ(x) = δ(ux) ∈ δ(U) ⊆ U
and uδ(x) ∈ uR ⊆ U . Therefore δ(u)x ∈ U . This means that I(U) is a ∆-
ideal of R. If A is a ∆-ideal of R that is contained in U , then AR ⊆ A ⊆ U
and hence A ⊆ I(U).
Lemma 13. Let U be a Lie ∆-ideal of R. If U is an associative subring
of R, then [U, U ] = 0 or U contains a nonzero ∆-ideal of R.
24 Lie and Jordan structures
Proof. Assume that x ∈ R and [U, U ] 6= 0. Then [u, v] 6= 0 for some
u, v ∈ U and
[u, vx] = u(vx) − (vx)u = (uv − vu)x + v(ux − xu).
Since [u, x], [u, vx] ∈ U and v[u, x] ∈ U , we deduce that [u, v]x ∈ U . This
means that [u, v] ∈ I(U). In view of Lemma 12, I(U) is a nonzero ∆-ideal
of R that is contained in U .
Proposition 2. If U is a Lie ∆-ideal of R, then [U, U ] = 0 or there
exists a nonzero ∆-ideal IU of R such that [IU , R] ⊆ U .
Proof. By Lemma 3 of [7],
T (U) = {t ∈ R | [t, R] ⊆ U}
is both a Lie ideal and an associative subring of R and U ⊆ T (U).
Moreover, for δ ∈ ∆, we have
[δ(t), R] + [t, δ(R)] = δ([t, R]) ⊆ δ(U) ⊆ U
and so [δ(t), R] ⊆ U . Hence T (U) is ∆-stable. If [U, U ] 6= 0, then, by
Lemmas 12 and 13,
IU = I(T (U)) ⊆ T (U)
is a nonzero ∆-ideal of R such that [IU , R] ⊆ U .
Lemma 14. Let U be a Lie ∆-ideal of a ring R. If [U, U ] = 0, then the
centralizer CR(U) is a Lie ∆-ideal and an associative subring of R.
Proof. Is immediately.
We extend Theorem 1.3 of [9] in the following
Proposition 3. Let R be a ∆-simple ring of characteristic 2. If U is a
Lie ∆-ideal of R, then one of the following holds:
(1) [R, R] ⊆ U ,
(2) U ⊆ Z(R),
(3) R contains a subfield P such that U ⊆ P and [P, R] ⊆ P .
O. D. Artemovych, M. P. Lukashenko 25
Proof. If [U, U ] 6= 0, then [R, R] ⊆ U by Proposition 2. Therefore we
assume that [U, U ] = 0. By Lemma 14, CR(U) is a Lie ∆-ideal and an
associative subring of R such that U ⊆ CR(U).
a) If CR(U) is non-commutative, then CR(U) = R by Lemma 13.
Hence U ⊆ Z(R).
b) Now assume that the centralizer CR(U) is commutative. If c ∈
CR(U) and x ∈ R, then
c2 ∈ CR(U) and [c2, x] = [[c, x], x] = 2c[c, x] = 0.
This gives that c2 ∈ Z(R). By Theorem 2 of [22], Z(R) is a field. As a
consequence, c2 (and so c) is invertible in CR(U). Hence CR(U) is a field.
Corollary 3. Let R be a ∆-simple ring. If U is a Lie ∆-ideal of R, then
one of the following holds:
(1) [R, R] ⊆ U ,
(2) U ⊆ Z(R),
(3) char R = 2 and R contains a subfield P such that U ⊆ P and
[P, R] ⊆ P .
4. Jordan properties
Lemma 15. Let R be a ∆-simple ring of characteristic 6= 2, U its proper
Jordan ∆-ideal and a ∈ U . If [a, R] ⊆ U , then a = 0.
Proof. Let us x, y ∈ R. Since [a, x] ∈ U and (a, x) ∈ U , we obtain that
2ax ∈ U and, as a consequence, ax ∈ U and (ax, y) ∈ U . Moreover, from
axy ∈ U it follows that yax ∈ U . This means that RaR ⊆ U . Since
d(a) ∈ U for any d ∈ ∆, in view of [21, Lemma 1.1] we obtain that
∞∑
k=1
∑
δ1,...,δk∈∆
(m1,...,mk)∈N
k
Rδm1
1 . . . δmk
k (a)R
is a proper ∆-ideal of R that is contained in U . Hence a = 0.
Remark 1. Let R be a 2-torsion-free ring, U its Jordan ∆-ideal. If ∆
contains all inner derivations of R, then U is an ideal of R.
26 Lie and Jordan structures
In fact, we have
2xa = [a, x] + (a, x) ∈ U
for any a, b, x ∈ U and so xa ∈ U . By the same argument, we can conclude
that ax ∈ U .
Proof of Theorem 1.
(1) (⇐) If A is a nonzero proper ∆-ideal of a ring R, then AJ is a
nonzero proper ∆-ideal of RJ , a contradiction.
(⇒) Let U be a proper Jordan ∆-ideal of R, a, b ∈ U and x ∈ R. By
Lemma 1 of [7], [(a, b), x] ∈ U , and, by Lemma 15, we see that
(a, b) = 0. (4.4)
In particular, 2a2 = 0 and, as a consequence, a2 = 0 and 2axa =
(a, (a, x)) = 0. It follows that axa = 0. Since
0 = (a + b)x(a + b) = axb + bxa
and
0 = (b, (a, x)) = b(ax + xa) + (ax + xa)b = bax + bxa + axb + xba,
we deduce that bax + xab = 0. But ab = −ba and so bax − xba = 0. This
means that ba ∈ Z(R). Then (RabR)2 = 0. Since
I =
∞∑
k=1
∑
a,b∈U, δ1,...,δk∈∆
(m1,...,mk)∈N
k
Raδm1
1 . . . δmk
k (b)R
is a ∆-ideal of R that is a sum of nilpotent ideals, we obtain that I = 0.
Therefore
0 = (b, x)a = (bx + xb)a = bxa + xba = 2bxa.
We conclude that URU = 0. From (RUR)2 = 0 and δ(RUR) ⊆ RUR for
any δ ∈ ∆ it holds that U = 0.
(2) (⇐) If A, B are ∆-ideals of R such that AB = 0, then (BA)2 = 0
and so BA is a Jordan ideal of R satisfying the condition
(BA, BA) = 0.
O. D. Artemovych, M. P. Lukashenko 27
Thus the condition (4.4) is true for U = BA. As in the proof of the part
(1), we obtain that BA = 0. Then AJ , BJ are ∆-ideals of a Jordan ring
RJ such that
(AJ , BJ) = 0.
Hence A = 0 or B = 0.
(⇒) Let a1, a2 ∈ A and x, y ∈ R. Suppose that RJ is not ∆-prime
and therefore there exist nonzero Jordan ∆-ideals A, B of R such that
(A, B) = 0.
By the same reasons as above, we conclude that A ∩ B = 0. Then, by
Lemma 1 of [7], we have [(a1, a2), x] ∈ A and hence
[(a1, a2), x] ± ((a1, a2), x) ∈ A.
Therefore x(a1, a2)y ∈ A. Thus R contains ∆-ideals R(A, A)R ⊆ A and
R(B, B)R ⊆ B such that
R(A, A)R(B, B)R ⊆ A ∩ B = 0.
Hence (A, A) = 0 or (B, B) = 0 and this leads to a contradiction.
(3) (⇐) If A is a nonzero ∆-ideal of R such that A2 = 0, then AJ is
a nonzero ∆-ideal of the Jordan ring RJ such that
(AJ , AJ) = 0,
a contradiction.
(⇒) Suppose that R has a nonzero Jordan ∆-ideal U such that
(U, U) = 0.
Then the condition (4.4) is true for any a, b ∈ U . As in the proof of the
part (1), we obtain that U = 0.
�
If R is a ring, then on the set R we can to define a left Jordan
multiplication “〈−, −〉” by the rule
〈a, b〉 = 2ab
for any a, b ∈ R. Then the equalities
〈〈〈a, a〉, b〉, a〉 = 〈〈a, a〉, 〈b, a〉〉 and 〈〈a, b〉, a〉 = 〈a, 〈b, a〉〉
28 Lie and Jordan structures
are true and hence
RlJ = (R, +, 〈−, −〉)
is a non-commutative Jordan ring (which is called a left Jordan ring
associated with an associative ring R). It is clear that, for commutative
ring R, we have
RJ = RlJ .
If A is an additive subgroup of R that 〈a, r〉, 〈r, a〉 ∈ A for any a ∈ A and
r ∈ R, then A is called an ideal of RlJ . If δ ∈ ∆ and a, b ∈ R, then
δ(〈a, b〉) = δ(2ab) = 2δ(a)b + 2aδ(b) = 〈δ(a), b〉 + 〈a, δ(b)〉
and therefore δ ∈ Der(RlJ). By the other hand, if δ ∈ Der(RlJ), then
2δ(ab) = δ(〈a, b〉) = 〈δ(a), b〉 + 〈a, δ(b)〉 = 2(δ(a)b + aδ(b)).
If R is a 2-torsion-free ring, then δ ∈ Der R. Similarly, as in Theorem 1,
we can prove the following
Proposition 4. For a 2-torsion-free ring R the following conditions are
true:
(1) R is a ∆-simple ring if and only if RlJ is a ∆-simple Jordan ring,
(2) R is a ∆-prime ring if and only if RlJ is a ∆-prime Jordan ring,
(3) R is a ∆-semiprime ring if and only if RlJ is a ∆-semiprime Jordan
ring.
5. Proofs
The next lemma in the prime case is contained in [18, Lemma 7].
Lemma 16 ([2, Lemma 1.7]). Let R be a ring. If [[R, R], [R, R]] = 0,
then the commutator ideal C(R) is nil.
Corollary 4. If R is a non-commutative ∆-semiprime ring, then [R, R]
is non-commutative.
Proof of Theorem 2.
(1) It is clear that a ring R is non-commutative. If A is a nonzero
proper ∆-ideal of R, then AL is a nonzero proper ∆-ideal of RL. Therefore
A ⊆ Z(R) and, as a consequence, A · C(R) = 0.
O. D. Artemovych, M. P. Lukashenko 29
(2) Suppose that a ∆-simple ring R is non-commutative and U is
its nonzero proper Lie ∆-ideal. By Proposition 2, [U, U ] = 0. Then, by
Lemma 11, U ⊆ Z(R). Hence the quotient ring RL/Z(R) is ∆̂-simple.
(3) Let A be a nonzero ∆-ideal of R such that A2 = 0. Then AL is a
nonzero ∆-ideal of a Lie ring RL and, moreover,
[AL, AL] = 0.
By Lemma 11, A ⊆ Z(R) and hence A · C(R) = 0.
(4) Suppose that R is non-commutative. Let A be a nonzero Lie
∆-ideal of R such that [A, A] = 0. Then, by Lemma 11, A ⊆ Z(R) and,
as a consequence, the Lie ring RL/Z(R) is ∆̂-semiprime.
(5) Let A, B be nonzero ∆-ideals of R such that AB = 0. Obviously,
[A, B] ⊆ Z(R). Then A ⊆ Z(R) or B ⊆ Z(R).
(6) Assume that R is non-commutative and A, B are nonzero Lie
∆-ideals of R such that
[A, B] = 0.
Then A ∩ B ⊆ Z(R). Since A ∩ B ⊆ ann C(R) in a ∆-prime ring R, we
have that the intersection A ∩ B = 0 is zero. If T (A) = R (see proof
of Proposition 2), then [R, R] ⊆ A and B ⊆ CR([R, R]). By Lemma 9,
B ⊆ Z(R). So we assume that T (A) 6= R. If [T (A), T (A)] = 0, then
[A, A] = 0 and, by Lemma 11, A ⊆ Z(R). Suppose that [T (A), T (A)] 6= 0.
By Lemma 13, T (A) contains a nonzero ∆-ideal I of R. Since
[I, B] ⊆ A ∩ B = 0,
we conclude that B ⊆ Z(R) by Lemma 7.
The map
∂a : R ∋ x 7→ [a, x] ∈ R
is called an inner derivation of a ring R induced by a ∈ R. The set IDer R
of all inner derivations of R is a Lie ring. Every prime Lie ring is primary
Lie.
Lemma 17. There is the Lie ring isomorphism
IDer R ∋ ∂a 7→ a + Z(R) ∈ RL/Z(R).
Proof. Evident.
Corollary 5. Let R be a ring. Then the following statements hold:
30 Lie and Jordan structures
(1) IDer R is a simple Lie ring if and only if RL/Z(R) is a simple Lie
ring,
(2) IDer R is a prime Lie ring if and only if RL/Z(R) is a prime Lie
ring,
(3) IDer R is a semiprime Lie ring if and only if RL/Z(R) is a semipri-
me Lie ring,
(4) IDer R is a primary Lie ring if and only if RL/Z(R) is a primary
Lie ring.
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Contact information
O. D. Artemovych Institute of Mathematics
Cracow University of Technology
ul. Warszawska 24
Cracow 31-155 POLAND
E-Mail(s): artemo@usk.pk.edu.pl
M. P. Lukashenko Faculty of Mathematics and Informatics
PreCarpathian National University of Vasyl Ste-
fanyk
Shevchenko St 57
Ivano-Frankivsk 76025 UKRAINE
E-Mail(s): bilochka.90@mail.ru
Received by the editors: 22.01.2015
and in final form 22.03.2015.
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