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

 

Minimal generating sets for free modules


Authors: L. M. Bruning and W. G. Leavitt
Journal: Proc. Amer. Math. Soc. 27 (1971), 441-445
MSC: Primary 16.40
Erratum: Proc. Amer. Math. Soc. 31 (1972), 638-638.
MathSciNet review: 0274498
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Abstract: Let $ R$ be a ring admitting a free module with generating set shorter than the length of a basis. If $ n$ is the shortest basis among all such modules and $ m$ the length of its shortest generating set then $ n = m + 1$ and every free module with basis of length $ \geqq m + 1$ has a generating set of length $ m$. If $ R$ has module type $ (h,k)$ then $ m = h$, that is an $ R$-module with basis of length $ u < h$ not only has all bases of length $ u$ but also has no generating set of length $ < u$. The integer $ m$ together with the module type define a new ring invariant which satisfies many of the properties of the module type.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0274498-8
PII: S 0002-9939(1971)0274498-8
Keywords: Generators of free modules, rank of a module, module type
Article copyright: © Copyright 1971 American Mathematical Society