Rings with finite essential socle

Pardo, José; Asensio, Pedro

doi:10.1090/S0002-9939-97-03747-7

Rings with finite essential socle
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by José L. Gómez Pardo and Pedro A. Guil Asensio

Proc. Amer. Math. Soc. 125 (1997), 971-977

DOI: https://doi.org/10.1090/S0002-9939-97-03747-7

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Abstract:

Let $R$ be a ring such that every direct summand of the injective envelope $E=E(R_R)$ has an essential finitely generated projective submodule. We show that, if the cardinal of the set of isomorphism classes of simple right $R$-modules is no larger than that of the isomorphism classes of minimal right ideals, then $R_R$ cogenerates the simple right $R$-modules and has finite essential socle. This extends Osofsky’s theorem which asserts that a right injective cogenerator ring has finite essential right socle. It follows from our result that if $R_R$ is a CS cogenerator, then $R_R$ is already an injective cogenerator and, more generally, that if $R_R$ is CS and cogenerates the simple right $R$-modules, then it has finite essential socle. We show with an example that in the latter case $R_R$ need not be an injective cogenerator.

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