A generalization of reversible rings

Communications of the Korean Mathematical Society, 2010

P. M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for a, b ∈ R. Commutative rings and reduced rings are reversible. In this paper, we extend the reversible condition of a ring as follows: Let R be a ring and α an endomorphism of R, we say that R is right (resp., left) α-shifting if whenever aα(b) = 0 (resp., α(a)b = 0) for a, b ∈ R, bα(a) = 0 (resp., α(b)a = 0); and the ring R is called α-shifting if it is both left and right α-shifting. We investigate characterizations of αshifting rings and their related properties, including the trivial extension, Jordan extension and Dorroh extension. In particular, it is shown that for an automorphism α of a ring R, R is right (resp., left) α-shifting if and only if Q(R) is right (resp., left)ᾱ-shifting, whenever there exists the classical right quotient ring Q(R) of R.

Journal of Pure and Applied Algebra, 2002

We determine the precise relationships among three ring-theoretic conditions: duo, reversible, and symmetric. The conditions are also studied for rings without unity, and the e ects of adjunction of unity are considered.

2014

We discuss the condition that the centre of a ring is an ideal. We also show that some classical commutativity results of Jacobson and Herstein have elementary proofs under the added assumption that the centre is an ideal.

2020

Regarding the question of how idempotent elements affect reversible property of rings, we study a version of reversibility depending on idempotents. In this perspective, we introduce {\it right} (resp., {\it left}) {\it $e$-reversible rings}. We show that this concept is not left-right symmetric. Basic properties of right $e$-reversibility in a ring are provided. Among others it is proved that if $R$ is a semiprime ring, then $R$ is right $e$-reversible if and only if it is right $e$-reduced if and only if it is $e$-symmetric if and only if it is right $e$-semicommutative. Also, for a right $e$-reversible ring $R$, $R$ is a prime ring if and only if it is a domain. It is shown that the class of right $e$-reversible rings is strictly between that of $e$-symmetric rings and right $e$-semicommutative rings.

Communications in Algebra, 2019

Our purpose in this paper is to show that certain subsets, defined by commutativity conditions involving derivations or endomorphisms, coincide with the center in prime rings. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.

Miskolc Mathematical Notes

In this note, we give a generalization for the class of *-IFP rings. Moreover, we introduce *-reversible and *-reflexive *-rings, which represent the involutive versions of reversible and reflexive rings and expose their properties. Nevertheless, the relation between these rings and those without involution are indicated. Moreover, a nontrivial generalization for *-reflexive *-rings is given. Finally, in *-reversible *-rings it is shown that each nilpotent element is *nilpotent and Köthe's conjecture has a strong affirmative solution.

Hacettepe Journal of Mathematics and Statistics, 2012

Let R be a ring with identity. We introduce a class of rings which is a generalization of reduced rings. A ring R is called central rigid if for any a, b ∈ R, a 2 b = 0 implies ab belongs to the center of R. Since every reduced ring is central rigid, we study sufficient conditions for central rigid rings to be reduced. We prove that some results of reduced rings can be extended to central rigid rings for this general setting, in particular, it is shown that every reduced ring is central rigid, every central rigid ring is central reversible, central semicommutative, 2-primal, abelian and so directly finite.

2011

The paper deals with �-reversible rings and their relationships with well known �-symmetric, �-Armendariz and �-semicommutative rings. We consider a skew version of some classes of rings with respect to a ring endomorphism�.

Bulletin of the Australian Mathematical Society, 2006

Let G be an arbitrary finite group, R be a finite associative ring with identity and RG be the group ring. We show that ℤ2Q8 is the minimal reversible group ring which is not symmetric, and we also characterise the finite rings R for which RQ8 is reversible. The first result extends a result of Gutan and Kisielewicz which shows that ℤ2Q8 is the minimal reversible group algebra over a field which is not symmetric, and it answers a question raised by Marks for the group ring case.

Transactions of the American Mathematical Society, 1975

We study nonassociative rings R satisfying the conditions (1) (ab, c, d) + (a, b, [c, d]) = a(b, c, d) + (a, c, d)b for all a, b, c, d e R, and (2) (x, x, x) = 0 for all x S. R. We furthermore assume weakly characteristic not 2 and weakly characteristic not 3. As both (1) and (2) are consequences of the right alternative law, our rings are generalizations of right alternative rings. We show that rings satisfying (1) and (2) which are simple and have an idempotent + 0, # 1, are right alternative rings. We show by example that (x, e, e) may be nonzero. In general, R = R' + (R, e, e) (additive direct sum) where R' is a subring and (R, e, e) is a nilpotent ideal which commutes elementwise with R. We examine R' under the added assumption of Lie admissibility: (3) (a, b, c) + (Jb, c, a) + (c, a, b) = 0 for all a, b, cEJi, We generate the Peirce decomposition. If R' has no trivial ideals contained in its center, the table for the multiplication of the summands is associative, and the nucleus of R' contains R'Xq + R'qX. Without the assumption on ideals, the table for the multiplication need not be associative; however, if the multiplication is defined in the most obvious way to force an associative table, the new ring will still satisfy (1), (2), (3). 1. Introduction. We study nonassociative rings which satisfy the generalized right alternative law: