Triangular matrix rings of selfinjective rings

Abstract A module M is said to be generalized extending if for every submodule there exists a direct summand D of M containing N such that D/N is a singular module. In this note we prove that a ring R is right self-injective if and only if the triangular ring is right generalized extending. This answers a question which was raised in A. Akalan, G.F. Birkenmeier, A. Tercan, Characterizations of extending modules and -extending generalized triangular matrix rings, Commun. Algebra 40 (2012), 1069–1085.