ON Locally Semi Right

Abstract

A new class of associative rings with identity, namely locally right semi-involution regular (L.r.s.I.Reg.) rings, is introduced and systematically studied in this paper.  A ring is defined to be locally right semi-involution regular if, for every element (a), either (a) or (1-a) is a right semi-involution regular element. The study establishes the fundamental properties of this class of rings and examines its relationships with several well-known classes of rings, including locally involution-regular rings, involution-regular rings, von Neumann-regular rings, and clean rings. Several characterizations of locally right semi-involution regular rings are presented, together with illustrative examples that clarify the new concept. It is shown that every element satisfying the defining conditions is a von Neumann regular element under suitable assumptions, and equivalent criteria involving direct summands of annihilator ideals are established. Furthermore, the paper proves that this class of rings is preserved under surjective ring homomorphisms and quotient constructions. Additional structural properties are obtained for abelian locally right semi-involution regular rings, including decomposition results involving semi-involution regular and nilpotent elements. The influence of principal injectivity on these rings is also investigated, yielding sufficient conditions under which locally right semi-involution regular rings become locally involution regular or clean rings. Moreover, conditions ensuring the existence of idempotent ideals are derived. The results presented extend existing work on locally involution regular rings and provide a broader framework for studying generalized regularity conditions in ring theory. These findings contribute to a deeper understanding of the algebraic structure of rings and their interrelationships with classical concepts in modern abstract algebra.