Triangular representations of splitting rings

Abstract:

The term “splitting ring” refers to a nonsingular ring R such that for any right R-module M, the singular submodule of M is a direct summand of M. If R has zero socle, then R is shown to be isomorphic to a formal triangular matrix ring $\left ( {\begin {array}{*{20}{c}} A & 0 \\ B & C \\ \end {array} } \right )$, where A is a semiprime ring, C is a left and right artinian ring, and $_C{B_A}$ is a bimodule. Also, necessary and sufficient conditions are found for such a formal triangular matrix ring to be a splitting ring.