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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Flat and FP-injectivity


Author: Saroj Jain
Journal: Proc. Amer. Math. Soc. 41 (1973), 437-442
MSC: Primary 16A52
MathSciNet review: 0323828
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Abstract: One of the main results of this paper is the characterization of left FP-injective rings as rings over which every finitely presented right module is torsionless. The necessity of the torsionless condition (in fact reflexivity) previously has been observed for special classes of FP-injective rings (e.g., self-injective rings (Kato and McRae), right and left coherent and self FP-injective rings (Stenström)). Furthermore, if the ring is left coherent, it is equivalent to say that every finitely presented right module is reflexive. We investigate the condition (called IF) that every injective right module is flat and characterize this via FP-injectivity. IF-rings generalize regular rings and QF-rings, and under certain chain or homological conditions coincide with regular or QF-rings (e.g., $ wgl\dim R \leqq 1 \Rightarrow {\text{regular}}$, noetherian, or $ {\text{right perfect}} \Rightarrow {\text{QF)}}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0323828-9
PII: S 0002-9939(1973)0323828-9
Keywords: QF-ring, IF-ring, self-injective ring, FP-injective ring, coherent ring, perfect ring, regular ring, semiprime Goldie ring, right and left annulets, injective module, FP-injective module, flat module, projective module, torsionless module, reflexive module, finitely presented module
Article copyright: © Copyright 1973 American Mathematical Society