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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.

References [Enhancements On Off] (What's this?)

  • [1] P. M. Cohn, Some remarks on the invariant basis property, Topology 5 (1966), 215-228. MR 33 #5676. MR 0197511 (33:5676)
  • [2] W. G. Leavitt, The module type of a ring, Trans. Amer. Math. Soc. 103 (1962), 113-130. MR 24 #A2600. MR 0132764 (24:A2600)

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Keywords: Generators of free modules, rank of a module, module type
Article copyright: © Copyright 1971 American Mathematical Society

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