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.
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