## A consequence of injectivity for modules

Let $R$ be a ring with 1. Suppose $Q$ is a left unital $R$-module with the property that every short exact sequence $0 \rightarrow Q \rightarrow M_1 \rightarrow N \rightarrow 0$ splits and suppose $\psi : L \rightarrow M$ is an injective module homomorphism. Prove that for every module homomorphism $\phi : L \rightarrow Q$ there exists a homomorphism $\Phi : M \rightarrow Q$ such that $\phi = \Phi \circ \psi$.

We will take the lifting property of to be the definition of “injective module” to avoid circular reasoning; we showed in these two exercises that every module is contained in a module with the lifting property, and our proof of this previous exercise depends on Baer’s criterion, which in turn uses only the lifting property of injective modules.

Note that there exists (by this exercise) an injective module $A$ and an injective module homomorphism $\theta : Q \rightarrow A$. Note that the sequence $0 \rightarrow Q \rightarrow A \rightarrow A/Q \rightarrow 0$ is exact, and so splits. Thus $A \cong_R Q \oplus A/Q$. By this previous exercise, $Q$ is injective, so that (by our definition) every homomorphism $\phi : L \rightarrow Q$ lifts to a homomorphism $\Phi : M \rightarrow Q$.