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Proceedings of the American Mathematical Society

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A note on modules over a commutative regular ring


Author: Mark L. Teply
Journal: Proc. Amer. Math. Soc. 29 (1971), 267-268
MSC: Primary 13.40
DOI: https://doi.org/10.1090/S0002-9939-1971-0276214-2
MathSciNet review: 0276214
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Abstract: An example is given of a commutative, von Neumann regular ring R, which has a module A satisfying the following conditions: (1) $ T(A) = \{ a \in A\vert(0:a)$ is an essential ideal of R} is a cyclic R-module; (2) $ A/T(A)$ is a cyclic R-module; and (3) $ T(A)$ is not a direct summand of A. This answers in the negative a question raised by R. S. Pierce.


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

  • [1] J. S. Alin and S. E. Dickson, Goldie’s torsion theory and its derived functor, Pacific J. Math. 24 (1968), 195–203. MR 0227249
  • [2] R. S. Pierce, Modules over commutative regular rings, Memoirs of the American Mathematical Society, No. 70, American Mathematical Society, Providence, R.I., 1967. MR 0217056

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0276214-2
Keywords: Regular ring, torsion submodule, Boolean ring, direct summand, cyclic module
Article copyright: © Copyright 1971 American Mathematical Society