On Self-Injective Perfect Rings

Abstract

Harada calls a ring R right simple-injective if every R-homomor-phism with simple image from a right ideal of R to R is given by left multiplication by an element of R. In this paper we show that every left perfect, left and right simple-injective ring is quasi-Frobenius, extending a well known result of Osofsky on self-injective rings. It is also shown that if R is left perfect and right simple-injective, then R is quasi-Frobenius if and only if the second socle of R is countably generated as a left R-module, extending many recent results on self-injective rings. Examples are given to show that our results are non-trivial extensions of those on self-injective rings. A ring R is called quasi-Frobenius if R is left (and right) artinian and left (and right) self-injective. A well known result of Osofsky [15] asserts that a left perfect, left and right self-injective ring is quasi-Frobenius. It has been conjectured by Faith [9] that a left (or right) perfect, right self-injective ring is quasi-Frobenius. This conjecture remains open even for semiprimary rings. Throughout this paper all rings R considered are associative with unity and all modules are unitary R-modules. We write M R to indicate a right R-module. The socle of a module is denoted by soc(M). We write N ⊆ M (N ⊆ ess M) to mean that N is a submodule (essential) of M. For any subset X of R, l(X) and r(X) denote, respectively, the left and right annihilators of X in R. A ring R is called right Kasch if every simple right R-module is isomorphic to a minimal right ideal of R. The ring R is called right pseudo-Frobenius (a right PF-ring) if R R is an injective cogenerator in mod-R; equivalently if R is semiperfect, right self-injective and has an essential right socle. A ring R is called right principally injective if every R-morphism from a principal right ideal of R into R is given by left multiplication. In [14], a ring R is called a right generalized pseudo-Frobenius ring (a right GPF-ring) if R is semiperfect, right principally injective and has an essential right socle. We write J = J(R) for the Jacobson radical of the ring R. Following Fuller [10], if R is semiperfect with a basic set E of primitive idempotents, and if e, f ∈ E, we say that the pair (eR, Rf) is an i-pair if soc(eR) ∼ = f R/f J and soc(Rf) ∼ = Re/Je.

On second socles of finitely cogenerated injective modules

“…Other authors approach to the problem above by investigating the condition that the lefi R-module $Soc_{2}(R)$ (or $J/J^{2}$ ) is finitely (countably) generated. These results can be found in [6], [7] and [12].…”

A Look at the Faith Conjecture

“…& Note that Lemma 1 (5) shows that a right ideal T socR R is simple if and only if dim D T D 1X The next result shows that if dimP D 1 we can obtain the converse to (6) and (7) Proof. These are all right ideals by (6) and (7) of Lemma 1. If T T R is a right ideal, then T J because R is local.…”

“…If T T R is a right ideal, then T J because R is local. Since P R is simple, either P T or P T 0X In the ®rst case, T X È P for X D V by Lemma 1 (6). If P T 0Y we show that T socR R X If t v p P T then, for v 1 P VY v v 1 v pv 1 P P T 0X Thus v P l V VY and so t P l V V È P socR R X & Note that, under the hypotheses of Lemma 2, the proper (two-sided) ideals of…”

“…(5). If x v p P socR R , where vV 0Y then xR fvd pd j d P Dg xDX (6). It is routine that X È P is a right ideal.…”