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Proceedings of the American Mathematical Society

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

 

 

$ {\rm p.p.}$ rings and finitely generated flat ideals


Author: Søren Jøndrup
Journal: Proc. Amer. Math. Soc. 28 (1971), 431-435
MSC: Primary 16.20
MathSciNet review: 0277561
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Abstract: In this note all rings considered are associative with an identity element 1 and all modules are unital left modules. It is shown that a commutative ring $ R$ has principal ideals projective if and only if $ R[X]$ has the same property. Furthermore it is proved that a ring $ R$ has all $ n$-generated left ideals flat if and only if all $ n$-generated right ideals are flat. In the last part of this note we will prove the following results:

Fix $ n \geqq 1$. Then there exists a ring $ R$ such that all $ n$-generated left ideals are projective, in particular, flat, while there exists a nonflat $ (n + 1)$-generated left ideal.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0277561-0
Keywords: Spectrum of a commutative ring, support of module, p.p. ring, $ n$-fir
Article copyright: © Copyright 1971 American Mathematical Society