The Sum-Product Splitting Property and Injective Direct Sums of Modules Over Von Neumann Regular Rings
Year: 1981
Type: article
Abstract: Let ${({M_i})_{i \in I}}$ be a family of modules over a von Neumann regular ring. It is shown that for the splitness of the canonical inclusion ${ \oplus _{i \in I}}{M_i} \subset \prod \nolimits _{i \in I} {{M_i}}$ it is necessary and sufficient that there be a finite subset $I’$ of $I$ such that the restricted sum ${ \oplus _{i \in I\backslash I’}}{M_i}$ is semisimple with finitely many homogeneous components, all simple summands being finite dimensional over their endomorphism rings. This yields a characterization of those families of injectives whose direct sum is again injective.
Author: Birge Zimmermann-Huisgen
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2041056590