New Results on EP Elements in Rings with Involution

In this paper, we mainly give characterizations of EP elements in terms of equations. In addition, a related notion named a central EP element is defined and investigated. Finally, we focus on characterizations of a generalized EP element, i.e., [Formula: see text]-DMP element.

Nil-Clean Rings with Involution

A [Formula: see text]-ring [Formula: see text] is called a nil [Formula: see text]-clean ring if every element of [Formula: see text] is a sum of a projection and a nilpotent. Nil [Formula: see text]-clean rings are the [Formula: see text]-version of nil-clean rings introduced by Diesl. This paper is about the nil [Formula: see text]-clean property of rings with emphasis on matrix rings. We show that a [Formula: see text]-ring [Formula: see text] is nil [Formula: see text]-clean if and only if [Formula: see text] is nil and [Formula: see text] is nil [Formula: see text]-clean. For a 2-primal [Formula: see text]-ring [Formula: see text], with the induced involution given by[Formula: see text], the nil [Formula: see text]-clean property of [Formula: see text] is completely reduced to that of [Formula: see text]. Consequently, [Formula: see text] is not a nil [Formula: see text]-clean ring for [Formula: see text], and [Formula: see text] is a nil [Formula: see text]-clean ring if and only if [Formula: see text] is nil, [Formula: see text]is a Boolean ring and [Formula: see text] for all [Formula: see text].

Symmetric Reverse Gamma *-4-Centralizers on Semiprime Gamma Rings with Involution

Let M be a -ring with involution . In this paper , we will introduce the concept of symmetric left(right) reverse *-4-centralizer of M . Then, we proved that the 4-additive mapping T:MxMxMxMM is a reverse *-4-centralizer of M under certain conditions .

A Note on Pair of Left Centralizers in Prime Ring with Involution

The purpose of this paper is to study pair of left centralizers in prime rings with involution satisfying certain identities.