Rings in which certain right ideals are direct summands of annihilators

Abstract

This paper is a continuation of the study of the rings for which every principal right ideal (respectively, every right ideal) is a direct summand of a right annihilator initiated by Stanley S. Page and the author in [20,21].

Some results on quasi-Frobenius rings

“…The following result is known, see [6,Corollary 4.8] Recall that a ring R is called right GC2 [9], [15] if every right ideal that is isomorphic to R is itself a direct summand; a ring R is called a right Goldie ring [7] if it has the maximum condition on right annihilators and R R is finite dimensional.…”

“…For example, a ring R is called right n-injective [5] if every R-homomorphism from an n-generated right ideal of R to R extends to an endomorphism of R. A right 1-injective ring is also said to be right P -injective [5]. A ring R is said to be right f.g self-injective [1] if it is right n-injective for each positive integer n. A ring R is called right YJ-injective [10], [12] or right generalized principally injective (briefly right GP-injective) [3], [4] if, for any 0 = a ∈ R, there exists a positive integer n such that a n = 0 and any right R-homomorphism from a n R to R extends to an endomorphism of R. A ring R is called right JGP-injective [9] if, for any 0 = a ∈ J(R), there exists a positive integer n such that a n = 0 and any right R-homomorphism from a n R to R extends to an endomorphism of R. A ring R is called right mininjective [6] if every R-homomorphism from a minimal right ideal of R to R extends to an endomorphism of R. A ring R is called right AGPinjective [15] if, for any 0 = a ∈ R, there exist a positive integer n and a left ideal X a n such that a n = 0 and lr(a n ) = Ra n ⊕ X a n . It is easy to see that the following implications hold: In this paper, we shall give some new characterizations of quasi-Frobenius rings, some conditions will be given under which a right 2-injective (resp., mininjective, YJ-injective, AGP-injective, JGP-injective) ring is quasi-Frobenius.…”

Almost Principally Small Injective Rings

“…In [17], a module M is said to satisfy the generalized C2-condition (GC2) if, for any N ⊆ M and N ∼ = M , N is a summand of M . …”

“…Many of the results on right P -injective rings were obtained for the two classes of right AP -injective rings and right AGP -injective rings. In [17], Zhou continued the study of left AP -injective rings and left AGP -injective rings with various chain conditions.…”

A Generalization of Simple-Injective Rings

“…A ring R is called right general principally injective (briefly right GP -injective) [3] if, for any 0 = a ∈ R, there exists a positive integer n such that a n = 0 and any right R-homomorphism from a n R to R extends to an endomorphism of R. A ring R is called right JGP-injective [15] if for any 0 = a ∈ J(R), there exists a positive integer n such that a n = 0 and any right R-homomorphism from a n R to R extends to an endomorphism of R. A ring R is called right MGP-injective [19,20] if, for any 0 = a ∈ R, there exists a positive integer n such that a n = 0 and any right R-monomorphism from a n R to R extends to an endomorphism of R. A ring R is called right AGP -injective [12,17] if for any 0 = a ∈ R, there exists a positive integer n such that a n = 0 and Ra n is a direct summand of l(r(a n )).…”