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Vladikavkaz Mathematical Journal 2021, Volume 23, Issue 2, P. 70-77
YAK 512.55
DOI 10.46698/d4945-5026-4001-v
A NOTE ON SEMIDERIVATIONS IN PRIME RINGS AND C*-ALGEBRAS#
M. A. Raza1 and N. Rehman2
1 Department of Mathematics, Faculty of Science & Arts-Rabigh, King Abdulaziz University, Jeddah 21589, Kingdom of Saudi Arabia;
2 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, Uttar Pradesh, India [email protected]; [email protected]; [email protected]
Abstract. Let R be a prime ring with the extended centroid C and the Matrindale quotient ring Q. An additive mapping F : R ^ R is called a semiderivation associated with a mapping G : R ^ R, whenever F(xy) = F(x)G(y) + xF(y) = F(x)y + G(x)F(y) and F(G(x)) = G(F(x)) holds for all x,y £ R .In this manuscript, we investigate and describe the structure of a prime ring R which satisfies F(xm o yn) £ Z(R) for all x,y £ R, where m,n £ Z+ and F : R ^ R is a semiderivation with an automorphism £ of R. Further, as an application of our ring theoretic results, we discussed the nature of C*-algebras. To be more specific, we obtain for any primitive C*-algebra A. If an anti-automorphism Z : A ^ A satisfies the relation (xn)z + xn* £ Z(A) for every x,y £ A, then A is C* — W4-algebra, i. e., A satisfies the standard identity W4(a1, a2, a3, a4) = 0 for all a1,a2,a3,a4 £ A.
Key words: prime ring, automorphism, semiderivation. Mathematical Subject Classification (2010): 16W25, 16N60.
For citation: Raza, M. A. and Rehman, N. A Note on Semiderivations in Prime Rings and C*-Algebras, Vladikavkaz Math. J., 2021, vol. 23, no. 2, pp. 70-77. DOI: 10.46698/d4945-5026-4001-v.
1. Introduction
Throughout the paper unless otherwise stated, R is the prime ring with centre Z(R), Q is the Martindale quotient ring of R and C is the extended centroid R (for further details see [1]). For given € R, the symbol [x,y] and x o y stands for the commutator and anti-commutator of x and y defined as xy — yx and xy+yx, respectively. We also note that a ring R is said to be a prime ring if aR6 = {0} implies that either a = 0 or b = 0. For any subsets A and B of R, [A, B] stands for the additive subgroup generated by [a, b] with a € A and b € B. Also, an additive subgroup L of R is said to be Lie ideal of R if [u, r] € L for all u € L and r € R. A mapping g : R ^ R is said to be commuting (resp. centralizing) on a subset S of R if [g(x),x] = 0 (resp. [g(x),x] € Z(R)) for all x € S. An additive mapping D : R R is called a derivation on R, if D(xy) = D(x)y + xD(y) holds for all x, y € R.
In [2], Bergen introduced the notion of semiderivation. An additive mapping F : R — R is called a semiderivation associated with a mapping G : R - R, whenever
F (xy) = F (x)G (y) + xF (y) = F (x)y + G (x)F (y)
#For the second author, this research is supported by the National Board of Higher Mathematics (NBHM), India, Grant № 02011/16/2020 NBHM (R. P.) R & D II/ 7786. © 2021 Raza, M. A. and Rehman, N.
and F(G(x)) = G(F(x)) holds for all x,y € R. For G = 1R, the identity map on R, F is clearly a derivation. Breser [3] proved that the only semiderivations of prime rings are ordinary derivations and mappings of the form F(x) = y(x — G(x)), where y € C and G is an endomorphism.
Let us briefly recall the motivation behind this study. In [4], Posner studied the centralizing derivations of prime rings and proved that if R is a prime ring and D is a non-zero derivation of R such that [D(x),x] € Z(R), for all x € R, then R is commutative. This result due to Posner was then extended to Lie ideals by Lanski [5]. In [6], Daif and Bell showed that a semiprime ring R must be commutative if it admits a derivation D such that either D([x,y])-[x,y] = 0 for all x,y € R or D([x, y]) + [x, y] = 0 for all x,y € R. In 2002, Ashraf and Rehman [7] obtained the same conclusion if the commutator is replaced by an anti-commutator which stated that if a prime ring R admits a derivation D such that D(x) o D(y) = x o y for all x, y € R, then R is commutative. In [8], Herstein proved that a ring R is commutative if it has no nonzero nilpotent ideal and there is a fixed integer n > 1 such that (xy)n = xnyn for all x,y € R. In [9], Bell proved that a prime ring R with nonzero center, for which char(R) = 0 or char(R) > n, where n > 1, must be commutative if it admits a nonzero derivation D such that D([xn,y] - [x,yn]) € Z(R) for all x,y € R. Further, Ali et al. [10] showed that if R be a 2-torsion free semiprime ring and it admits a derivation D such that D(xm o yn) € Z(R) for all x,y € R, then R is commutative (for additional associated results [11-14]).
On the other hand, recently Haung [15] proved that a prime ring R satisfies s4, the standard identity in four variables if char(R) > n + 1 or char(R) = 0 and F(x)n = 0 holds, where x € L, a noncentral Lie ideal of R and F is a semiderivation associated with an automorphism £ of R.
Given the above discussions, we investigate and describe the structure of a ring R which satisfies certain identities involving automorphisms and semi-derivations. Also, we discuss the nature of C*-algebras. To be more specific, we obtain the following theorems:
Theorem 1.1. Let R be a prime ring of char(R) = 2 and m, n € Z+. If an automorphism Z of R satisfies (xm o yn)z € Z(R) for all x, y € R, then R satisfies s4, the standard identity in four variables.
Theorem 1.2. Let R be a prime ring of char(R) = 2 and m,n € Z+. If a semiderivation F associated with an automorphism £ such that F(xm o yn) € Z(R). Then R satisfies s4, the standard identity in four variables.
Theorem 1.3. Let A be a primitive C*-algebra and m,n € Z+. If an automorphism £ : A ^ A satisfies the relation (xm o yn)z € Z(A) for all x, y € A, then A is C* - W4-algebra.
Theorem 1.4. Let A be a primitive C*-algebra and n € Z+. If an anti-automorphism Z : A ^ A satisfies the relation (xn)z + xn* € Z(A) for every x,y € A, then A is C* - W4-a\gebra.
Before proving our main results, we fix some notions which are required for the exposition of our main results. An automorphism £ is called Q-inner if there exists an invertible element q € Q such that £(x) = qxq-1 for all x € R. Also, the standard identity s4 in four variables is defined as follows:
2. Preliminaries
where (- 1)M is a sign of permutation / of the symmetric group of degree 4. Further we mention the following results which are crucial in developing the proof of our main theorem.
Fact 2.1. Let R be a prime ring and I a two sided ideal of R. Then I, R, Q satisfy the same generalized polynomial identities with coefficients in Q (see [16]). Furthermore, I, R and Q satisfy the same generalized polynomial identities with automorphisms (see [17, Theorem 1]).
Fact 2.2. Let R be a prime ring with extended centroid C. Then the following conditions are equivalent:
(i) dimC RC ^ 4.
(ii) R satisfies s4, the standard identity in four variables.
(iii) R is commutative or R embeds in M2(F) for F a field.
(iv) R is algebraic of bounded degree 2 over C.
(v) R satisfies [[x2,y], [x,y]] = 0.
Fact 2.3. Let R be a prime ring and L a be non-central Lie ideal of R. If char(R) = 2, by [18, Lemma 1] there exists a nonzero ideal I of R such that 0 = [I, R] C L. If char(R) = 2 and dimCRC > 4, i.e., char(R) = 2 and R does not satisfy s4, then by [19, Theorem 13] there exists a nonzero ideal I of R such that 0 = [I, R] C L. Thus if either char(R) = 2 or R does not satisfy s4, then we may conclude that there exists a nonzero ideal I of R such that [I, I] C L.
3. Main Results
Proposition 3.1. Let R be a dense subring of End(VD) and Z : R R be an automorphism of R. If R satisfies ([x1,x2] o [y1,y2])z € Z(R) for all x1,x2,y1,y2 € R, then either dim(VD) ^ 2 or Z is an identity map on End(VD).
< First assume that VD be a right vector space over a division ring D. Let End(VD) the ring of D-linear transformations on VD. Thus in view of classical Jacobson Theorem [20, Isomorphism Theorem, p. 79], we have sz = PsP-1 for every s € End(VD), where Z is an automorphism of End(VD) and P is an invertible semi-linear transformation. Hence, for all v € V, Z € D, P(v^>) = (Pv)ZGiven by the hypotheses, we obtain
0= [[x!,x2]C[y1,y2]Z + [y1,y2]Z[x1,x2]Z, z] = [P[x1, x^!, y2] P-1 + P[y1, yfc, x2]P-1,z]
for every x1,x2,y1,y2, z € End(VD). Let us assume that v and P-1v are D-dependent for every v € V .In view of [21, Lemma 1], we find that P -1v = vx, where x € D and v € V. Hence, for all s € End(VD), P-1(sv) = svx and sv = P(svx) = P(s(vx)) = PsP-1(v) = szv for all s € End(VD), v € V. Therefore, we find that (sz-s)V = (0) for every s € End(vD). Hence, sz = s for every s € End(VD). This shows that Z is an identity map on End(VD), as required.
Thus, there exists v € V such that v and P-1v are linearly D-independent. Firstly, we assume that dim(VD) ^ 4. Then we may take w, Pv € V such that {w, v, Pv, P-1 v} is D-independent. Let x, y € End(VD) such that
x1v = 0, x1P-1v = 0, x1w = v, y1P-1v = 0, zv = 0; x2v = w, x2P-1v = v, y1v = v, y2P-1v = v, zPv = w.
We notice that [x1,x2]P-1v = 0, [y1,y2]P-1v = v, [x1,x2]v = v and hence, our assumption yields
0
P [x1,x2][y1,y2]P-1 + P [y1,y2][x1,x2]P ,Z
v = -w,
a contradiction, implying that dim(VD) ^ 3.
Secondly, we assume that dim(VD) — 3. Take Pv € V such that {v, Pv, P v} is D-independent and then {v, Pv, P-1 v} forms a D-basis of V. If P(v + P-1v + Pv) € vD and P(P-1v + Pv) € vD, then Pv, P(P-1 v + Pv) € vD and then v, P-1v + Pv € P-1(vD) = P-1(v)Z-1(D) = P-1vD, contradicting the fact that {v, P-1v + Pv} is D-independent. Therefore, one can pick p € {0,1} such that u = pv + P-1v + Pv and Pu / vD. Write Pu = va + P-1v^ + Pvy, where a, 7 € D and 7 both are not zero. By density of theorem, there exist x1,x2,y1,y2, z € End(VD) such that
x1v = 0, x2v = Pv, y1v = v, y2v = v, zv = 0; x1P-1v = v, x2P-1v = 0, y1P-1v = 0, y2P-1v = v, zP-1v = v; x1P v = u, x2P v = 0, y1Pv = v, y2Pv = v, zPv = u.
That is x1u = (p + 1)v + P-1v + Pv, x2u = Pv, y1u = (p + 1)v and y2u = —uy. Therefore, we can see that [x1,x2]P-1v = —P-1v, [y1,y2]P-1v = v, [x1,x2]v = u, [y1,y2]P-1v = 0. Also, zPu = v^ + uy. As y are not both zero and v, u are D-dependent, so it is easy to see that zPu = 0. Thus in all, we see that
a contradiction, implying that dim(VD) ^ 2. >
Theorem 3.1. Let R be a non-commutative prime ring of characteristic different from two and Z be an automorphism of R. If R satisfies ([x1,x2] o [y1,y2])z € Z(R) for all x1,x2,y1,y2 € R, then R satisfies s4, the standard identity in four variables.
< Firstly, we assume that Z is an inner automorphism of R, i.e., sz = psp-1 for every s € R .As Z is the non-identity map, so p / C. Then
is a non-trivial generalized polynomial identity (GPI) of R and hence of Q as well. By Martindale's theorem [22], Q is isomorphic to dense subring of the ring of linear transformations of a vector space V over D, where D is a finite dimensional division ring over C. By Proposition 3.1, we have dim(VD) ^ 2. Thus it follows that either Q = D or Q = M2(D), the ring of 2x2 matrices over D. More generally, we assume that Q = (D), for k ^ 2.
If C is finite, then D is field by Wedderburn's theorem. On the other hand, if C infinite, let F be the algebraic closure of C, therefore by the Van der monde determinant argument, we see that Q ®cF satisfies the generalized polynomial identity ^(r) = 0. Moreover, Q ®cF =
(D) ®C F = (D ®C F) = Mt(F), for some t ^ 1. Considering Proposition 3.1 and the fact that Q is not commutative, we assert that t = 2, yields the required conclusion.
Secondly, we assume that Z is an outer automorphism. By [17, Theorem 1], Q and hence R satisfy [[x1,x2]z[y1,y2]z + [y1,y2]z[x1,x2]z, z] = 0. As xz, yz-word degree < char(R), then by [23, Theorem 3], R satisfies [[x'1,x/2][y/1 ,y2] + [yi,y2][x1,x'2],z] = 0. That is, R is a polynomial identity (PI) ring. Thus, R and Mt(F) satisfy the same polynomial identities [24, Lemma 1], i.e., for each x/1,x2,y/1 ,y2,z € Mt(F), [[x1,x2][y/1 ,y2] + [y'i,y2][x1, x^], z] = 0. Take k ^ 3 and ej, the usual unit matrix. Therefore, for x = e23, y = e32, z = e11, s = e12, we get a contradiction 0 = [[x^xZny',y2] + [y',y2][xi,x//],z] = [[en,e12][e23,e32]+ [e23,e32][en,e12], [e23,e32]] = e12 = 0. Hence t = 2, i.e., R satisfies s4, the standard identity in four variables. This completes the proof. >
0
P [x1,x2][y1,y2]P-1 + P [y1,y2][x1,x2]P-1,z )v
jv = —zP u.
^(r) = P [x1,x2][y1,y2]P-1 + P [y1,y2][x1,x2]P-1,z
< Proof of Theorem 1.1. We are given that (xm o yn)z € Z(R) for every x,y € R. Let S1 = {rm : r € R} and S2 = {rn : r € R} be the additive subgroups. It implies that (a o b)z € Z(R) for all a € S1, b € S2. In view of [25, Main theorem], and since char(R) = 2, either S1 have a non-central Lie ideal L of R or rm € Z(R) for all r € R. The latter case concludes R to be commutative. Similarly, assume that there exists a Lie ideal L2 C Z(R) such that L2 C S2. Moreover, in view of Fact 2.3, there exist 1 and 12 nonzero two-sided ideals of R such that 0 = [1^ R] C L and 0 = [12, R] C L2. Also, R is non-commutative as L1, L2 are non-central Lie ideal of R. Therefore (x o y)z € Z(R) for all x € [11,11], y € [12,12]. Since 11, 12 and R satisfy the same differential identities (see [24, Theorem 3]), so we have (xoy)z € Z(R) for all x, y € [R, R]. By Theorem 3.1, we get the required result. >
Using the same technique as used in Theorem 1.1 and Theorem 3.1, we can write in view of above result
Theorem 3.2. Let R be a non-commutative prime ring of characteristic different from two and F be a non-zero semiderivation associated with an automorphism £ of R. If R satisfies F([x1,x2] o [y1,y2]) € Z(R) for all x1,x2,y1,y2 € R, then R satisfies s4, the standard identity in four variables.
< First we note that if £ is an identity map on R, then F is not more than a derivation. In view of previous discussion, we have nothing to prove. Hence, we proceed by assuming that £ is not an identity map on R. Hence in view of Bresar [3], F(x) = y(x-x^) for all x € R, where 0 = y € C. Thus by our hypothesis we can write y([x1,x2] o [y1,y2] - ([x1,x2] o [y1,y2])^) € Z(R) which can be rewritten as y(([x1, x2] o [y1, y2])lR - ([x1, x2] o [y1, y2])^) € Z(R), where Ir is the identity map on R. It is well known that if £ is an automorphism of R, then £ + klR (k is an any integer) is also an automorphism on R. Thus, we set £ - Ir = Z- Therefore, the last relation can be written as y([x1,x2] o [y1,y2])z € Z(R) for all x1,x2,y1,y2 € R. Since 0 = y € C, the above identity reduces to ([x1,x2] o [y1,y2])z € Z(R) for all x1,x2,y1,y2 € R and hence in view of Theorem 3.1, we get the desired conclusion. >
Proof of Theorem 1.2. We are given that F(xm o yn) € Z(R) for every x,y € R. Let S1 = {rm : r € R} and S2 = {rn : r € R} be the additive subgroups. It is easy to see that F(x o y) € Z(R) for each x € S1, y € S2. Since char(R) = 2 and by main theorem of [25], we have either rm € Z(R) for every r € R or S1 contains a non-central Lie ideal L1 of R. The first case concludes that R to be commutative. Similarly, assume that there exists a Lie ideal L2 C Z(R) such that L2 C S2. According to Fact 2.3, there exist nonzero two-sided ideals I1 and 12 of R such that 0 = [I1, R] C L1 and 0 = [12, R] C L2. Since L1, L2 are non-central Lie ideal of R, so R is non-commutative. Hence, F(x o y) € Z(R) for all x € [11,11],y € [12,12]. Since 11, 12 and R satisfy the same differential identities (see [24, Theorem 3]), so we have F(xoy) € Z(R) for all x,y € [R,R]. Applying Theorem 3.2, we are done.
Corollary 3.1. Let R be a prime ring of characteristic different from two, m be fixed positive integer and F be a nonzero semiderivation associated with an automorphism £ of R. If F (xm) € Z(R) for all x, y € R, then R satisfies s4, the standard identity in four variables.
Corollary 3.2. Let R be a prime ring of characteristic not two. If R admits an automorphism Z of R such that (xn)z € Z(R) for all x € R, then R satisfies s4, the standard identity in four variables.
Theorem 3.3. Let R be a prime ring of characteristic not two. If R admits an automorphism Z of R such that (xn)z+xn € Z(R) for all x € R, then R satisfies s4, the standard identity in four variables.
< It is well known that if Z is an automorphism of R, then Z + klR (k is an any integer) is also an automorphism on R. We have given that (xn)z + xn € Z (R) for all x € R which can be rewritten as (xn)z + (xn)lR € Z(R), where Ir is the identity map on R. Thus, we set Z — Ir = Therefore, the last relation can be written as (xn)^ € Z(R) for all x € R and hence by Corollary 3.2 we have done. >
4. Result Based on C*-Algebras
A Banach algebra is a linear associate algebra which, as a vector space, is a Banach space with norm || ■ || satisfying the multiplicative inequality; ||xy|| ^ ||x||||y|| for all x and y in A .A Banach algebra A is a PI-algebra if and only if there exists n € N and a polynomial q € Wn, q = 0, such that q(x1,x2,... ,xn) = 0 for all x1,x2,... ,xn € A, where Wn is the set of all complex polynomials in n non-commuting variables. An involution on an algebra A is a map x i—> x* of stf onto such that the following conditions are hold: (i) ixy)* = y*x*, (ii) (a;*)* = x, and (iii) (x + Ay)* = x* + Xy* for all x, y € stf and A € C the field of complex number, where A is the conjugate of A. Of course the prototypical example of an involution on a Banach algebra is the adjoint operation on B(H), the set of bounded linear operators on Hilbert space H. Another important example is complex conjugation on C(X), the set of all continuous complex valued functions on X, a compact Hausdroff space defined as f*(x) := f(x).
An algebra equipped with an involution is called a *-algebra or algebra with involution. A Banach *-algebra is a Banach algebra A together with an isometric involution ||x*|| = ||x|| for all x € A .A Banach *-algebra is called a C *-algebra A if || x x 11 — 11 x ||2 for all x € A. A C*-algebra A is primitive if its zero ideal is primitive, that is, if A has a faithful nonzero irreducible representation. Let Wn denote the standard polynomial of degree n in n non-commuting variables, Wn = sign (a)aCT(1)aCT(2) ••• aCT(n), where Sn is the set of
all permutations of {1,2,3, ••• ,n} and sign(a) = ±1 for a even (odd) (see [26, 27] and references therein). An algebra A is said to be an C*-W„-algebra if Wn(a1,a2, ■ ■ ■ ,an) = 0 for each choice of elements a1, a2, ■ ■ ■ , an € A .In particular, an algebra is C * — W4-algebra if it satisfies the standard identity W4(a1, a2, a3, a4) = 0 for all a1,a2,a3,a4 € A. Moreover, an algebra is C* — W2-algebra if and only if it is commutative, i.e., a C* — W2-algebra is commutative if it satisfies the standard identity W2(a1,a2) = 0 for all a1,a2 € A. Many researcher discussed Gelfand's theory for Banach algebra and C*-algebra namely, Banach-W2n-algebra and C* — W2n-algebra. Throughout the present section, C*-algebras are assumed to be nonunital unless indicated otherwise.
< Proof of Theorem 1.3. We have given that Z : A — A is an automorphism of A and A is a primitive C*-algebra such that (xm o yn)z € Z(A) for all x,y € A. Therefore, A is prime by [28, Theorem 5.4.5] because A is primitive C*-algebra. Hence, A is a prime ring since A is a prime C*-algebra. By application of Theorem 1.1 get the required conclusion, thereby proving the theorem. >
< Proof of Theorem 1.4. We have (xn)z + xn* € Z(A) for all x € A. Replace x* for x, to get (xn*)z + xn € Z(A) for all x € A. Now, a map n : A — A by xn = x*^ for every x € A .It is easy to see that (xy)n = xn yn for all x, y € A, that is, n is an automorphism of A and hence we find that (xn)n + xn € Z(A) for every x € A. Therefore, A is prime by [28, Theorem 5.4.5] because A primitive C*-algebra. Hence, A is a prime ring since A is a prime C*-algebra. Application of Theorem 3.3 yields the required conclusion. >
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Received September 7, 2020
Mohd Arif Raza
Department of Mathematics,
Faculty of Science & Arts-Rabigh,
King Abdulaziz University,
Jeddah 21589, Kingdom of Saudi Arabia,
Associate Professor
E-mail: [email protected]
https://orcid.org/0000-0001-6799-8969
Nadeem ur Rehman Department of Mathematics, Aligarh Muslim University, Aligarh 202002, Uttar Pradesh, India, Professor
E-mail: [email protected], [email protected]
https://orcid.org/0000-0003-3955-7941
Владикавказский математический журнал 2021, Том 23, Выпуск 2, С. 70-77
ПОЛУДИФФЕРЕНЦИРОВАНИЯ В ПЕРВИЧНЫХ КОЛЬЦАХ
Раза М. А.1, Рехман Н.2
1 Университет короля Абдул-Азиза, Саудовская Аравия, 21589, Джидда;
2 Алигархский мусульманский университет, Индия, 202002, Алигарх [email protected]; [email protected], [email protected]
Аннотация. Пусть R — первичное кольцо с расширенным центроидом C и с фактор-кольцо Мат-риндейла Q. Аддитивное отображение F : R ^ R называют полупроизводной, ассоциированной с G : R ^ R, если F(xy) = F(x)G(y) + xF(y) = F(x)y + G(x)F(y) и F(G(ж)) = G(F(x)) для всех x, y £ R. В этой работе мы исследуем и описываем строение первичных колец R, удовлетворяющих условию F(xm о yn) £ Z(R) для всех x,y £ R, где m,n £ Z+ и F : R ^ R — полупроизоводная с автоморфизмом £ кольца R. Далее, в качестве приложения нашего теоретико-кольцевого результата мы обсуждаем природу C*-алгебр. Точнее, для любой примитивной C*-алгебры A. Точнее, для любой примитивной C*-алгебры A получаем следующее. Если антиизоморфизм Z : A ^ A удовлетворяет соотношению (xn)z + xn* £ Z(A) для всех x, y £ A, то A служит C* — W^-алгеброй, т. е., A удовлетворяет стандартному тождеству W4(a1, a2, a3, a4) = 0 for all a1,a2,a3,a4 £ A.
Ключевые слова: первичное кольцо, автоморфизм, полупроизводная.
Mathematical Subject Classification (2010): 16W25, 16N60.
Образец цитирования: Raza, M. A. and Rehman, N. A Note on Semiderivations in Prime Rings and C*-Algebras // Владикавк. мат. журн.—2021.—Т. 23, № 2.—C. 70-77 (in English). DOI: 10.46698/d4945-5026-4001-v.
