Complete pure injectivity and endomorphism rings

Gómez Pardo, J. L.; Dung, Nguyen V.; Wisbauer, R.

doi:10.1090/S0002-9939-1993-1137232-X

Complete pure injectivity and endomorphism rings
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by J. L. Gómez Pardo, Nguyen V. Dung and R. Wisbauer

Proc. Amer. Math. Soc. 118 (1993), 1029-1034

DOI: https://doi.org/10.1090/S0002-9939-1993-1137232-X

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Abstract:

It is shown that if $M$ is a finitely presented completely pure injective object in a locally finitely generated Grothendieck category ${\mathbf {C}}$ such that $S = {\operatorname {End} _{\mathbf {C}}}M$ is von Neumann regular, then $S$ is semisimple. This is a generalized version of a well-known theorem of Osofsky, which includes also a result of Damiano on PCI-rings. As an application, we obtain a characterization of right hereditary rings with finitely presented injective hull.

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