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Proceedings of the American Mathematical Society

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

 

 

On semilocal $ {\rm OP}$-rings


Author: Yukitoshi Hinohara
Journal: Proc. Amer. Math. Soc. 32 (1972), 16-20
MSC: Primary 13.95
DOI: https://doi.org/10.1090/S0002-9939-1972-0289502-1
MathSciNet review: 0289502
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Abstract: The notion of OP-rings was introduced by D. Lissner. A commutative ring R is called an OP-ring if, for any $ n \geqq 2$, any vector of $ {R^n}$ is an outer product of $ n - 1$ vectors of $ {R^n}$. Recently J. Towber proved that any local ring is an OP-ring if and only if the maximal ideal is generated by two elements. The main result in the present paper is a generalization to semilocal rings of the above theorem proved by Towber for local rings. The author's argument does not rely on Towber's theorem however, and so provides a new and very elementary proof of that result.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0289502-1
Keywords: OP-ring, unimodular vector, outer product, semilocal ring, local ring, Jacobson radical, principal ideal, principal ideal ring, special principal ideal ring, projective module
Article copyright: © Copyright 1972 American Mathematical Society