A Generalization of Semiregular and Almost Principally Injective Rings

In this article, we call a ring R right almost I-semiregular for an ideal I of R if for any a is an element of R, there exists a left R-module decomposition l(R)r(R)(a) = P circle plus Q such that P subset of Ra and Q boolean AND Ra subset of I, where l and r are the left and right annihilators, respectively. This generalizes the right almost principally injective rings defined by Page and Zhou, I-semiregular rings defined by Nicholson and Yousif, and right generalized semiregular rings defined by Xiao and Tong. We prove that R is I-semiregular if and only if for any a is an element of R, there exists a decomposition l(R)r(R)(a) = P circle plus Q, where P = Re subset of Ra for some e(2) = e is an element of R and Q boolean AND Ra is an element of I. Among the results for right almost I-semiregular rings, we show that if I is the left socle Soc((R)R) or the right singular ideal Z(R(R)) or the ideal Z((R)R) boolean AND delta((R)R), where delta((R)R) is the intersection of essential maximal left ideals of R, then R being right almost I-semiregular implies that R is right almost J-semiregular for the Jacobson radical J of R. We show that delta(l)(eRe) = e delta((R)R)e for any idempotent e of R satisfying ReR = R and, for such an idempotent, R being right almost delta((R)R)-semiregular implies that eRe is right almost delta(l)(eRe)-semiregular.