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.