The sum-product splitting property and injective direct sums of modules over von Neumann regular rings

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.