Semiregular, Semiperfect and Perfect Rings Relative to an Ideal

Abstract

Let I be an ideal of a ring R. Consider the following conditions on R:1.If X is a finitely generated submodule of a finitely generated projective module P , then X = A ⊕ B where A is a summand of P and B ⊆ P • I.2. If X is a submodule of a finitely generated projective module P , then X = A ⊕ B where A is a summand of P and B ⊆ P • I.3. If X is a submodule of a projective module P , then X = A ⊕ B where A is a summand of P and B ⊆ P • I.When I is the Jacobson radical J(R) of R, these conditions characterize semiregular rings, semiperfect rings and right perfect rings, respectively.In this paper we completely characterize these conditions for the cases when I is the right singular ideal, or the right socle, or the intersection of any two of the three ideals.As applications, structure theorems are obtained for right CEP-rings R with J(R) 2 = 0 and for QF-rings R with J(R) 2 = 0.All rings R are associative and have an identity, unless otherwise specified, and modules are unitary right modules over R. For an Rmodule M , J(M ), Z(M ) and Soc (M ) are the Jacobson radical, the singular submodule and the socle of M , respectively.We use Z r , Z l , S r and S l to indicate the right singular ideal, the left singular ideal, the right socle and the left socle of R, respectively.1. I-Semiregular rings.The following lemma has been observed in [21, Lemma 1.1] when K is a principal right ideal of R.Lemma 1.1.Let I be an ideal of the ring R. The following conditions are equivalent for a right ideal K of R: