Semiprime Rings Research Articles

On Certain Identities with Automorphisms on Lie Ideals in Prime and Semiprime Rings

Let R be a prime ring of characteristic different from 2, Z(R) its center, L a Lie ideal of R, and m, n, s, t ≥ 1 fixed integers with t ≤ m + n + s. Suppose that α is a non-trivial automorphism of R and let Φ(x, y) = [x, y]t – [x, y]m [α([x, y]),[x, y]]n [x, y]s. Thus, (a) if Φ(u, v) = 0 for any u, v ∈ L, then L ⊆ Z(R); (b) if Φ(u, v) ∈ Z(R) for any u, v ∈ L, then either L ⊆ Z(R) or R satisfies s4, the standard identity of degree 4. We also extend the results to semiprime rings.

Generalized derivations with nilpotent, power-central, and invertible values in prime and semiprime rings

Let R be a ring, Q its symmetric Martindale quotient ring, C its extended centroid, I a nonzero ideal of R and F a generalized derivation with associated non-zero derivation d of R, and fixed integers. Let be a non-zero multilinear polynomial over C in t non-commuting variables, be any subset of R and . We prove the following results:If R is prime and for all , then is central valued on R.If R is prime and , for all , then is power central valued on R, unless .If R is semiprime and for all , then , for any and , that is there exists a central idempotent element such that , d vanishes identically on eQ and is central valued on .If R is semiprime and is zero or invertible in R, for all , then either R is a division ring or it is the ring of 2 × 2 matrices over a division ring, unless when , for any and .If R is prime and I is a non-zero right ideal of R such that and , for all , then is an identity on I.Let R be prime and I a non-zero right ideal of R such that and , for all . If there exists such that , then either is power central valued on R or is an identity on I, unless .

On generalized derivations in semiprime rings involving anticommutator

Let $$\mathfrak {R}$$ be a prime ring with characteristic different from 2 and m, n, k be fixed positive integers. In this paper we study the case when $$\mathfrak {R}$$ admits a generalized derivation $$\mathscr {F}$$ with associated derivation $$\mathfrak {D}$$ such that $$(i)~(\mathscr {F}(x)\circ \mathscr {F}(y))^k=\mathscr {F}(x\circ _ky)~(ii)~ \mathscr {F}(x)\circ _m \mathscr {F}(y) =(\mathscr {F}(x\circ y))^n~(iii)~(\mathscr {F}\left( x\right) \circ \mathscr {F}\left( y\right) )^m =(\mathscr {F}(x\circ y))^n,$$ for all $$x,y \in \mathscr {I}$$ , where $$ \mathscr {I}$$ is a non-zero ideal of $$\mathfrak {R}$$ . Moreover, we also examine the case when $$\mathfrak {R}$$ is a semiprime ring.

Annihilators and extensions of idempotent-generated ideals

We define a ring R to be right -Baer if the right annihilator of a cyclic projective right R-module in R is generated by an idempotent. This class of rings generalizes the class of right p.q.-Baer rings. We also define a ring R to be right I-extending if each ideal generated by an idempotent is essential, as a right R-module, in a right ideal generated by an idempotent. It is shown that the two conditions are equivalent in a semiprime ring. From the definition of a right -Baer ring, we define a generalization of a prime ring which is equivalent to the condition that all nonzero cyclic projective right R-modules are faithful. Such a ring is called right I-prime. For a semiprime ring, we show the existence of a -Baer hull. We also provide some results about the p.q.-Baer hull and when it is equal to the -Baer hull. Polynomial and formal power series rings are studied with respect to the right -Baer condition. In general, a formal power series ring over one indeterminate in which its base ring is right p.q.-Baer ring is not necessarily right p.q.-Baer. However, if the base ring is right -Baer then the formal power series ring over one indeterminate is right -Baer. The last section is devoted to matrix extensions of right -Baer rings. A characterization of when a 2-by-2 generalized upper triangular matrix ring is right -Baer is given. The last major theorem is a decomposition of a -Baer ring R, satisfying a mild finiteness condition, into a direct sum of rings where A is a direct sum of right I-prime rings; and B is a generalized triangular matrix ring with right I-prime rings down the main diagonal. Examples illustrating and delimiting our results are provided.

Certain basic functional identities of semiprime rings

The goal of the article is to discuss certain basic functional identities of semiprime rings. Specifically, the following result is proved: Let R be a semiprime ring with extended centroid C and let be additive maps, where denotes the maximal left ring of quotients of R. Suppose that for all . Then there exist and additive maps such that and for all . Moreover, some related functional identities are also characterized.

Generalized biderivations on prime and semiprime rings

Generalized biderivations on prime and semiprime rings

Differential Inverse Power Series Rings with Quasi-Armendariz-like Condition

A generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of differential inverse power series type (or simply, [Formula: see text]-quasi-Armendariz), is introduced and studied. It is shown that the [Formula: see text]-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. Various classes of non-semiprime [Formula: see text]-quasi-Armendariz rings are provided, and a number of properties of this generalization are established. Some characterizations for the differential inverse power series ring [Formula: see text] to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP are concluded, where δ is a derivation on the ring R. Finally, miscellaneous examples to illustrate and delimit the theory are given.

Commuting multiplicative generalized derivations on Lie ideals of semiprime rings

Let R be a semiprime ring with characteristic different from two and L a noncentral square-closed Lie ideal of R. Suppose that R admits a multiplicative generalized derivation (F,d) satisfying d(L) ⊆ L. In the present paper, we shall prove that d is commuting on L if one of the following conditions holds: (i) F([x,y]) = ∓ [x,d(y)]; (ii) F(x ∘ y) = ∓ (x ∘ d(y)); (iii) F([x,y]) = ∓ (F(y)x); (iv) F([x,y]) = ∓ (F(x)y); (v) F(x ∘ y) = ∓ (F(y)x); (vi) F(x ∘ y) = ∓ (F(x)y) for all x,y ∈ L.

Centralizers and Jordan triple derivations of semiprime rings

Let R be a semiprime ring with extended centroid C and with maximal left ring of quotients . An additive map is called a Jordan triple derivation if for all . In 1957, Herstein proved that a Jordan triple derivation, which is also a Jordan derivation, of a noncommutative prime ring of characteristic 2, must be a derivation. In 1989, Brešar proved that any Jordan triple derivation of a 2-torsion free semiprime ring is a derivation. In the article, we give a complete characterization of Jordan triple derivations of arbitrary semiprime rings. To get such a characterization we first show that, in some sense, an additive map satisfying for all can be realized as a centralizer with only an exceptional case that and R is commutative.

Orthogonality of two left and right generalized derivations on ideals in semiprime rings

In this paper, we present some results concerning orthogonality of l-generalized derivations and r-generalized derivations on ideals in semiprime rings, we also study the connections between orthogonality and some properties of the composition of a l-generalized derivation and a r-generalized derivation. These results are related to results of Bresar and Vukman (Rad Mat 5(2):237–246, 1989), that extend a theorem by Posner (Proc Am Math Soc 8:1093–1100, 1957) about products of derivations on prime rings.

On a Functional Equation Related to Two-Sided Centralizers

AbstractThe main aim of this manuscript is to prove the following result. Let n > 2 be a fixed integer and R be a k-torsion free semiprime ring with identity, where k ∈ {2, n - 1, n}. Let us assume that for the additive mapping T : R→ R 3T(xn) = T(x)xn-1+ xT(xn-2)x + xn-1T(x), x ∈R, is also fulfilled. Then T is a two-sided centralizer.

On certain equations in semiprime rings and standard operator algebras

Abstract The purpose of this paper is to prove the following result which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let {\mathcal{L}(X)} be the algebra of all bounded linear operators of X into itself and let {\mathcal{A}(X)\subset\mathcal{L}(X)} be a standard operator algebra. Suppose there exist linear mappings {\mathcal{H},\mathcal{G}\colon\mathcal{A(X)}\to\mathcal{L(X)}} satisfying the relations \displaystyle\mathcal{H}(\mathcal{A}^{m+n})=\mathcal{H}(\mathcal{A}^{m})% \mathcal{A}^{n}+\mathcal{A}^{m}\mathcal{G}(\mathcal{A}^{n}), \displaystyle\mathcal{G}(\mathcal{A}^{m+n})=\mathcal{G}(\mathcal{A}^{m})% \mathcal{A}^{n}+\mathcal{A}^{m}\mathcal{H}(\mathcal{A}^{n}) for all {\mathcal{A}\in\mathcal{A(X)}} and some fixed integers {m,n\geq 1} . Then there exists {\mathcal{B}\in\mathcal{L(X)}} , such that {\mathcal{H(A)}=\mathcal{AB}-\mathcal{BA}} for all {\mathcal{A}\in\mathcal{F(X)}} , where {\mathcal{F(X)}} denotes the ideal of all finite rank operators in {\mathcal{L}(X)} , and {\mathcal{H}(\mathcal{A}^{m})=\mathcal{A}^{m}\mathcal{B}-\mathcal{B}\mathcal{A% }^{m}} for all {\mathcal{A}\in\mathcal{A(X)}} .

Permuting Jordan Left Tri – Derivations On Prime and semiprime rings

Let be a 2 and 3 – torsion free prime ring then if admits a non-zero Jordan left tri- derivation , then is commutative ,also we give some properties of permuting left tri - derivations.

Multiplicative (Generalized) Reverse Derivations on Semiprime Ring

Let R be a semiprime ring. A mapping F : R → R (not necessarily additive) is called a multiplicative (generalized) reverse derivation if there exists a map  d : R → R (not necessarily a derivation nor an additive map) such that F(xy) = F(y)x + yd(x) for all x, y є R. In this paper we investigate some identities involving multiplicative (generalized) reverse derivation and prove some theorems in which we characterize these mappings.

Centrally-extended generalized *-derivations on rings with involution

The aim of the present paper is to establish some results concerning centrally-extended *-derivations and centrally-extended generalized *-derivations. Also, we investigate the commutativity of semiprime rings with such maps.

A note on multiplicative (generalized)-skew derivation on semiprime rings

ABSTRACTIn this article, we study multiplicative (generalized)-skew derivation G and multiplicative left centralizer H satisfying certain conditions in semiprime rings. Moreover, some examples are given to demonstrate that the semiprimeness imposed on the hypotheses of various theorems is essential.

Projections of Finite Commutative Rings with Identity

Associative rings R and R′ are said to be lattice-isomorphic if their subring lattices L(R) and L(R′) are isomorphic. An isomorphism of the lattice L(R) onto the lattice L(R′) is called a projection (or a lattice isomorphism) of the ring R onto the ring R′. A ring R′ is called the projective image of a ring R. We study lattice isomorphisms of finite commutative rings with identity. The objective is to specify sufficient conditions subject to which rings under lattice homomorphisms preserve the following properties: to be a commutative ring, to be a ring with identity, to be decomposable into a direct sum of ideals. We look into the question about the projective image of the Jacobson radical of a ring. In the first part, the previously obtained results on projections of finite commutative semiprime rings are supplemented with new information. Lattice isomorphisms of finite commutative rings decomposable into direct sums of fields and nilpotent ideals are taken up in the second part. Rings definable by their subring lattices are exemplified. Projections of finite commutative rings decomposable into direct sums of Galois rings and nilpotent ideals are considered in the third part. It is proved that the presence in a ring of a direct summand definable by its subring lattice (i.e., the Galois ring GR(pn,m), where n > 1 and m > 1) leads to strong connections between the properties of R and R′.

On maximal commutative subrings of non-commutative rings

ABSTRACTWe observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.

Some differential identities on prime and semiprime rings and Banach algebras

In this manuscript, we discuss the behaviour and nature of generalized derivation \({\mathscr {G}}\) on a (semi-) prime ring \({\mathscr {R}}\) satisfying certain differential identities over \({\mathscr {I}}\), a nonzero ideal of \({\mathscr {R}}\). Moreover, we extend our purely ring theoretic result to a non-commutative Banach algebras and obtained some range inclusion results of continuous generalized derivations.

Ad-nilpotent Elements of Semiprime Rings with Involution

AbstractLet R be an n!-torsion free semiprime ring with involution * and with extended centroid C, where n > 1 is a positive integer. We characterize a ∊ K, the Lie algebra of skew elements in R, satisfying (ada)n = 0 on K. This generalizes both Martindale and Miers’ theorem and the theorem of Brox et al. In order to prove it we first prove that if a, b ∊ R satisfy (ada)n = adb on R, where either n is even or b = 0, then (a − λ)[(n+1)/2] = 0 for some λ ∊ C.