2 K, R, Goodearl and A. K, Boyle

by maximality of H. Consequently Z(C/H) = 0 and so H e L*(C),

whence K ^ H, Therefore A ^ K.

e

Conversely, let M e L*(C) such that A ^ M, and note that

K S M. Observing that M/A is singular, we see that M/K is

singular also, whence M/K ^ Z(C/K) = 0. Therefore M = K.

DEFINITION, Given any module A and any cardinal * (finite

or infinite), we use oA to denote the direct sum of ot copies

of A.

DEFINITION, A module A is subisomprphic to a module B,

written A B, provided A is isomorphic to a submodule of B,

THEQHEM 1,2, [2, Theorem] If A and B are infective modules

such that A B and B A, then A 2-B.

DEFINITION, For any ring R, we use 71(H) to denote the full

subcategory of Mod-R generated by all nonsingular infective right

R-modules, Clearly 7?(R) is an additive category with arbitrary

products and finite coproducts. The following proposition identifies

subobjects in 7KR).

PROPOSITION 1.5, Let C e 7?(R), and let A be any submodule

of C, Then the following conditions are equivalent:

(a) A

e

7J(R).

(b) A

e

L*(C).

(c) A is a closed submodule of C.

(d) A is a direct summand of C.

Proof, (c) * * (d) follows from the injectivity of C, and

(d) =* (a) is clear.