NONSINGULAR INJECTIVE MODULES 3
(a) = * (b): Since A is infective, we must have A@B = C for
some B. Then C/A is isomorphic to the nonsingular module B,
whence A e L*(C).
(b) = » (c): If A ^ B ^ C, then B/A is singular and so
B/A £ Z(C/A) = 0.
In view of l«3i we may identify the subobjects of any object
C e 7?(R) with the members of L*(C), which we refer to as "closed
submodules of CTt. Since we have frequent occasion to use the fact
that closed submodules of C are direct summands of C, we do not
always refer to 1*3•
DEFINITION. We use E(A) to denote the injective hull of a
PROPOSITION l.k. Let C e 7J(R), let A £ Cf and let K be
the ^-closure of A in C. Then K is uniquely determined by the
conditions K ^ C, K = E(A).
Proof. This follows directly from 1.1 and 1.3.
PROPOSITION 1.5. If A,B e 7?(R) and f e HonigU.B), then
ker f e L*(A) and fA
Proof. Since A/(ker f) is isomorphic to a submodule of the
nonsingular module B, we must have ker f e L*(A). Then
A s- (ker f)e(fA), whence fA
71(B) and consequently fA e L*(B).
Proposition 1,5 shows that any map f : A -*B in 7?(R) has a
kernel and image in the category 7?(R), and that they coincide with
the kernel and image of f in the category Mod-R. As a result,