All rings in this paper are associative with unit, all modules
are unital, and modules are right modules unless otherwise specified.
As a result, all endomorphism rings act on the left of their modules.
The letter R always denotes a ring. For basic terminology and
notation regarding nonsingular rings and modules, we refer the
reader to [**•]. In particular, we use ,fA ^ B" to mean that A is
a submodule of B, and " A ^ BH to mean that A is an essential
submodule of B#
DEFINITION. A closed submodule of a module C is a submodule
A which has no proper essential extensions inside C, i.e.,
A ^ B S C is satisfied only for B = A.
DEFINITION. An g-closed submodule of a module C is a
submodule A such that C/A is nonsingular. We use L*(C) to
denote the set of all S-closed submodules of C. Since any direct
product of nonsingular modules is nonsingular, it follows that L*(C)
is closed under arbitrary intersections. Given any A ^ C, we thus
have a smallest S-closed submodule of C containing A, called the
g-closure of A in C.
PROPOSITION 1.1. Let C be a nonsingular module, let A g C,
and let K be the g-closure of A in C. Then K is uniquely
determined by the conditions A 3 » K, K e L*(C).
Proof. By definition, K e L*(C). By Zorn's Lemma, C has a
submodule H which is maximal with respect to the property
A ^ H % C. If J/H = Z(C/H), then since C is nonsingular,
[4, Proposition 1.5] says that H % J. Now A ^ J and so J = H,