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• W1700570470 abstract "Harada calls a ring R right simple-injective if every R-homomorphism 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 MR 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 RR 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) ∼= fR/fJ and soc(Rf) ∼= Re/Je. Received by the editors April 24, 1995 and, in revised form, October 11, 1995. 1991 Mathematics Subject Classification. Primary 16D50, 16L30." @default.
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• W1700570470 date "1997-01-01" @default.
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• W1700570470 title "On perfect simple-injective rings" @default.
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• W1700570470 doi "https://doi.org/10.1090/s0002-9939-97-03678-2" @default.
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