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 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]**Carl Faith,*Rings with ascending condition on annihilators*, Nagoya Math. J.**27**(1966), 179–191. MR**0193107****[2]**Helmut Lenzing,*Direct sums of projective modules as direct summands of their direct product*, Comm. Algebra**4**(1976), no. 7, 681–691. MR**0404335****[3]**Birge Zimmermann-Huisgen,*The sum-product splitting property and injective direct sums of modules over von Neumann regular rings*, Proc. Amer. Math. Soc.**83**(1981), no. 2, 251–254. MR**624908**, 10.1090/S0002-9939-1981-0624908-5**[4]**Wolfgang Zimmermann,*Rein injektive direkte Summen von Moduln*, Comm. Algebra**5**(1977), no. 10, 1083–1117 (German). MR**0450327**

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

Article copyright:
© Copyright 1985
American Mathematical Society