e-Symmetric rings

Abstract

Let [Formula: see text] be a ring and [Formula: see text] an idempotent of [Formula: see text], [Formula: see text] is called an [Formula: see text]-symmetric ring if [Formula: see text] implies [Formula: see text] for all [Formula: see text]. Obviously, [Formula: see text] is a symmetric ring if and only if [Formula: see text] is a [Formula: see text]-symmetric ring. In this paper, we show that a ring [Formula: see text] is [Formula: see text]-symmetric if and only if [Formula: see text] is left semicentral and [Formula: see text] is symmetric. As an application, we show that a ring [Formula: see text] is left min-abel if and only if [Formula: see text] is [Formula: see text]-symmetric for each left minimal idempotent [Formula: see text] of [Formula: see text]. We also introduce the definition of strongly [Formula: see text]-symmetric ring and prove that [Formula: see text] is a strongly [Formula: see text]-symmetric ring if and only if [Formula: see text] and [Formula: see text] is a symmetric ring. Finally, we introduce [Formula: see text]-reduced ring and study some properties of it.

Semicommutativity of rings by the way of idempotents

“…In [13] e-symmetric rings and e-reduced rings are introduced and investigated. Let R be a ring and e 2 = e ∈ R. Then R is called e-symmetric if whenever abc = 0, then acbe = 0 for every a, b, c ∈ R. The ring R is right (resp.…”

“…left) e-reduced if ae = 0 (resp. ea = 0) for each nilpotent a ∈ R. By motivated these e-contexts, in that vein, in this section we will introduce and study the structures of right e-semicommutative rings generalizing e-symmetric rings and right e-reduced rings [13]. Throughout this paper, e denotes an idempotent element of a ring R which is under consideration.…”

“…The notion of an e-semicommutative ring is an extension of semicommutative rings as well as a generalization of e-reduced rings. Right e-reduced rings are e-symmetric by[13, Corollary 4.3]. We show that every e-symmetric ring is right e-semicommutative.…”

“…A ring R is symmetric [11] if abc = 0 implies acb = 0 for all a, b, c ∈ R. In [13] symmetric rings are generalized to e-symmetric rings. The ring R is called e-symmetric for e 2 = e ∈ R if abc = 0 implies acbe = 0 for all a, b, c ∈ R. Every symmetric ring is semicommutative.…”

Some properties of e -symmetric rings

“…In [11], Wei introduced generalized weakly symmetric rings, which further generalized the concept of symmetric rings. In [7], a ring R is called (strongly) e -symmetric if for any a, b, c ∈ R , abc = 0 implies (aceb = 0 ) acbe = 0 , where e ∈ E(R) . It is shown that a ring R is e -symmetric if and only if e is left semicentral and eRe is symmetric [7,Theorem 2.2].…”