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 module 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 e L*(B). 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,
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