Figure 3 . : SolF|S : with a subrepresentation $j 72S and morphisms
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can be completed to a commutative diagram as follows : commutes and H|s, = hand K|g, =k. In other words, every diagram of representations there exist H € Hompy(N, M1) and kK € Homp(N2, M2) such that the following diagram as a commutative diagram. Thus, Mj is an injective left R-module. PROOF. The lemma follows by completing the diagram Then, since E is an injective left R-module, there exists a map h : Mz — E. Now choose ho f : M, — E as an R-linear map. Then h and ho I complete the above diagram as a commutative diagram. Therefore, E —» E is an injective representation. Since My +t, Mpg is an injective representation, we can complete the above diagram as a commutative diagram as follows: PROOF. Consider the following diagram : References is an injective representation if and only if each M; is an injective left R- module and the representation is the direct sum of the following injective representations:
