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On Zimmermann-Huisgen's splitting theorem

Author: Victor Camillo
Journal: Proc. Amer. Math. Soc. 94 (1985), 206-208
MSC: Primary 16A64; Secondary 16A30, 16A52
MathSciNet review: 784163
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Abstract: This note is motivated by a paper of Birge Zimmermann-Huisgen, which in turn is motivated by a long sequence of papers--the first due to Faith--dealing with the question of when the canonical embedding of a direct sum of modules in the corresponding direct product splits. Zimmermann-Huisgen answered a question raised by previous authors by showing that if $ R$ is a von Neumann regular ring the only way this can happen is that, except for a finite number, the modules involved must each be semisimple with only a finite number of simple modules involved.

Based on a new, more elementary argument, we establish a necessary condition for the sum-product splitting over an arbitrary (associative) ring ft (with identity).

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

  • [1] C. F. Faith, Rings with ascending chain condition on annihilators, Nagoya Math. J. 27 (1966), 179-191. MR 0193107 (33:1328)
  • [2] H. Lenzing, Direct sums of modules as direct summands of their direct product, Comm. Algebra 4 (1976), 681-691. MR 0404335 (53:8137)
  • [3] B. Zimmermann-Huisgen, The sum-product splitting property and injective direct sums of modules over von Neumann regular rings, Proc. Amer. Math. Soc. 83 (1981), 251-254. MR 624908 (82k:16031)
  • [4] W. Zimmermann, Rein injektive direkte Summen von Moduln, Comm. Algebra 5 (1977), 1083-1117. MR 0450327 (56:8623)

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Article copyright: © Copyright 1985 American Mathematical Society

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