Extension of an $R$-homomorphism as a sum

Question

I want to solve this problem:

Let $M$ be an injective module over a ring $R$ with identity, and $f$, $g$, $h$ are elements of the ring of $R$-endomorphisms $\operatorname{End}(M)$ such that $\ker (g)∩\ker (h)⊆\ker (f)$. Verify that there are $g_1$ and $h_1$ in $\operatorname{End}(M)$ with $f=g_1\circ g+h_1\circ h$.

My try is to, first, define a subset $S=\{(g(x),h(x)): x∈M$} of $M⊕M$ which is easily seen to be a submodule of $M⊕M$, and define an $R$-homomorphism $k:S→M$ by the rule $k(g(x),h(x))=f(x)$. By the hypothesis, this $k$ is well-defined and could be extended to an $R$-homomorphism $k_1:M⊕M→M$ because the functor $\operatorname{Hom}(-,M)$ is, by injective-ness of $M$, exact.

Thanks for anybody helping me and completing the proof.

Answer 1

Take $g_1=k_1\circ \sigma_1$ and $h_1=k_1\circ \sigma_2$ where $\sigma_1,\sigma_2:M\to M\oplus M$, $\sigma_1(x)=(x,0)$ and $\sigma_2(y)=(0,y)$.