Minimal generating sets for free modules
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- by L. M. Bruning and W. G. Leavitt
- Proc. Amer. Math. Soc. 27 (1971), 441-445
- DOI: https://doi.org/10.1090/S0002-9939-1971-0274498-8
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Erratum: Proc. Amer. Math. Soc. 31 (1972), 638-638.
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
- P. M. Cohn, Some remarks on the invariant basis property, Topology 5 (1966), 215–228. MR 197511, DOI 10.1016/0040-9383(66)90006-1
- W. G. Leavitt, The module type of a ring, Trans. Amer. Math. Soc. 103 (1962), 113–130. MR 132764, DOI 10.1090/S0002-9947-1962-0132764-X
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 27 (1971), 441-445
- MSC: Primary 16.40
- DOI: https://doi.org/10.1090/S0002-9939-1971-0274498-8
- MathSciNet review: 0274498