The singular submodule of a finitely generated module splits off

A characterization is given of the finitely generated non-singular left i?-modules N such that Extβ (N, M) = 0 for every singular left i?-module M. As a corollary, the rings R, over which the singular submodule Z(A) is a direct summand of every finitely generated left i^-module A, are characterized. This characterization takes on a simplified form whenever R is commutative. An example is given to show that a ring R, over which the singular submodule Z(A) is a direct summand of every left ϋί-module A, need not be right semi-hereditary. In this paper, all rings R are assumed to be associative with an identity element, and, unless otherwise stated, all jβ-modules will be unitary left R-modules. A submodule B of an iϋ-module A is an essential submodule of A if B Π C Φ 0 for all nonzero submodules C of A. A left ideal