On Zimmermann-Huisgen's splitting theorem

Author:
Victor Camillo

Journal:
Proc. Amer. Math. Soc. **94** (1985), 206-208

MSC:
Primary 16A64; Secondary 16A30, 16A52

DOI:
https://doi.org/10.1090/S0002-9939-1985-0784163-6

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 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).

**[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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DOI:
https://doi.org/10.1090/S0002-9939-1985-0784163-6

Article copyright:
© Copyright 1985
American Mathematical Society