Reversible and reflexive properties for rings with involution

Advances in Pure Mathematics

Let R be a ring and () , S ≤ a strictly ordered monoid. In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce the notions of generalized power series reflexive and nil generalized power series reflexive, respectively. We obtain various necessary or sufficient conditions for a ring to be generalized power series reflexive and nil generalized power series reflexive. Examples are given to show that, nil generalized power series reflexive need not be generalized power series reflexive and vice versa, and nil generalized power series reflexive but not semicommutative are presented. We proved that, if R is a left APP-ring, then R is generalized power series reflexive, and R is nil generalized power series reflexive if and only if R/I is nil generalized power series reflexive. Moreover, we investigate ring extensions which have roles in ring theory.

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.

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.

International journal of pure and applied mathematics, 2015

I.N. Herstein as well as Susan Montgomery have defined and made many contributions to the algebraic structures of a ring R with involution *. An intrinsic algebraic subset of such a ring is the set of symmetric elements S = {a|a = a � }. A ring R with involution * is self-symmetric provided all its nontrivial ideals are symmetric. This paper will show that all self-symmetric rings with involution * are MP3 (Moore-Penrose 3) rings. In addition, some attempts will be made to characterize the n × n matrix ring, Mn(R) over a self-symmetric MP3 ring.

We study some properties related to zero divisors and reversibility in noncommutative rings.

Journal of Pure and Applied Algebra, 2003

A ring R is called reversible if ab = 0 implies ba = 0 for a; b ∈ R. We continue in this paper the study of reversible rings by Cohn [4]. We ÿrst consider properties and basic extensions of reversible rings and related concepts to reversible rings, including some kinds of examples needed in the process. We next show that polynomial rings over reversible rings need not be reversible, and sequentially argue about the reversibility of some kinds of polynomial rings. Moreover we prove that if R is a reduced ring then R[x]=(x n ) is a reversible ring, where (x n ) is the ideal generated by x n and n is a positive integer; and that for a right Ore ring R with Q its classical right quotient ring, R is reversible if and only if Q is reversible.

2018

We introduce a weakly symmetric ring which is a generalization of a symmetric ring and a strengthening of both a GWS ring and a weakly reversible ring, and investigate properties of the class of this kind of rings. A ring R is called weakly symmetric if for any a, b, c ∈ R, abc being nilpotent implies that Racrb is a nil left ideal of R for each r ∈ R. Examples are given to show that weakly symmetric rings need to be neither semicommutative nor symmetric. It is proved that the class of weakly symmetric rings lies also between those of 2-primal rings and directly finite rings. We show that for a nil ideal I of a ring R, R is weakly symmetric if and only if R/I is weakly symmetric. If R[x] is weakly symmetric, then R is weakly symmetric, and R[x] is weakly symmetric if and only if R[x;x−1] is weakly symmetric. We prove that a weakly symmetric ring which satisfies Köthe’s conjecture is exactly an NI ring. We also deal with some extensions of weakly symmetric rings such as a Nagata exte...

International Journal of Open Problems in Computer Science and Mathematics, 2017

In this paper we establish some commutativity criteria for a ring with involution (R, *) in which derivations satisfy certain algebraic identities. Some related results characterizing commutativity of prime rings have been discussed. Furthermore, we provide examples to show that the conditions imposed in the hypotheses of our results are necessary.

Proceedings - Mathematical Sciences, 2020

A ring R is called reversible if for a, b ∈ R, ab = 0 implies ba = 0. These rings play an important role in the study of noncommutative ring theory. Kafkas et al. (Algebra Discrete Math. 12 (2011) 72-84) generalized the notion of reversible rings to central reversible rings. In this paper, we extend the notion of central reversibility of rings to ring endomorphisms. We investigate various properties of these rings and answer relevant questions that arise naturally in the process of development of these rings, and as a consequence many new results related to central reversible rings are also obtained as corollaries to our results.

Pacific Journal of Mathematics, 1980

Osborn characterizes those associative rings with involutions in which each symmetric element is nilpotent or invertible. Analogous results are obtained for alternative rings. The restriction is further relaxed to require only that each symmetric element is nilpotent or some multiple is a symmetric idempotent.