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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Trivial extension of a ring with balanced condition


Author: Hideaki Sekiyama
Journal: Proc. Amer. Math. Soc. 77 (1979), 1-6
MSC: Primary 16A36
DOI: https://doi.org/10.1090/S0002-9939-1979-0539618-3
MathSciNet review: 539618
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Abstract: A ring R is called QF-1 if every faithful R-module is balanced. In this paper we study commutative QF-1 rings. It is shown that a commutative QF-1 ring is local if and only if it is uniform. It is well known that commutative artinian QF-1 rings are QF, but Osofsky has constructed a nonartinian nonnoetherian commutative injective cogenerator, so QF-1, ring which is a trivial extension of a valuation ring. It is shown that if a trivial extension of a valuation ring is QF-1, then it has a nonzero socle. Furthermore such rings become injective cogenerator rings under certain conditions.


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DOI: https://doi.org/10.1090/S0002-9939-1979-0539618-3
Keywords: QF-1 ring, double centralizer, trivial extension, valuation ring, injective cogenerator
Article copyright: © Copyright 1979 American Mathematical Society