A generalization of projective covers

Abstract

Let M be a left module over a ring R and I an ideal of R. We call (P , f ) a projective I -cover of M if f is an epimorphism from P to M, P is projective, Ker f ⊆ I P , and whenever P = Ker f + X, then there exists a summand Y of P in Ker f such that P = Y + X. This definition generalizes projective covers and projective δ-covers. Similar to semiregular and semiperfect rings, we characterize I -semiregular and I -semiperfect rings which are defined by Yousif and Zhou using projective I -covers. In particular, we consider certain ideals such as Z( R R), Soc( R R), δ( R R) and Z 2 ( R R).

Strong lifting splits

“…Then bT = 0 for some 0 ̸ = b ∈ R by the Kasch condition. Hence b ∈ l(a) = 0, a contradiction, and again we are done by (1).…”

“…On the other hand, N is a partial summand of N + X by [1,Lemma 3.2] because N is a partial summand of M. This means that N + X = S 1 + X where S 1 ⊆ ⊕ (N + X ) and S 1 ⊆ N. Since M = (N + X ) ⊕ Q , it follows that S 1 + Q ⊆ ⊕ M. Now write S = S 1 + Q . We have shown that S ⊆ ⊕ M, and clearly S = S 1 + Q ⊆ N + K .…”

“…If e 2 = e ∈ I then Re is summand-small by (1). To show that I is enabling, let a − e ∈ I where e 2 = e; we must find f 2 = f ∈ Ra such that a − f ∈ I.…”

Supplemented Modules Relative to an Ideal

“…Let N be a submodule of a module M . Recall that N is said to be DM in M (or N decomposes M ) if there is a direct summand D of M such that D ≤ N and M = D + X, whenever N + X = M for a submodule X of M (see [1,Definition 3.1]). Clearly, the following implications hold:…”

Weakly $I$-semiregular Rings and $I$-semiregular Rings

“…It is easy to see that I-semiregular rings are left-right symmetric. I-semiregular rings have been studied by many authors (see, for example [2,12,13,16,17,20]). By [12,Theorem 1.2] or [13,Theorem 28], we see that if R is left I-semiregular, then R/I is regular and idempotents can be lifted modulo I.…”