Show that the image or the kernel are submodule of R-module.

Question

Let $R$ be Commutative ring and $M$ be an $R$-module. Show that $im(H)$ or $ker(H)$ are submodule of $R$-module $M$, where $$H\in Hom(M,-)$$ First, I think by lemma:
If $M$ is an $R$-module and $N$ is a nonempty subset of $M$, then $N$ is an $R$-module of $M$ if and only if has $a_1m_1+a_2m_2$ $\in$ $N$ for all $m_1,m_2$ $\in$ $N$, and all $a_1,a_2$ $\in$ $R$.
But I couldn't completly prove this problem. I need some your help, please.

Answer 1

Let $y_1 \in im(H)$ and $y_2 \in im(H)$. so $y_1=H(x_1) , y_2=H(x_2)$. then $r_1y_1+r_2y_2 = r_1H(x_1)+r_2H(x_2)= H(r_1x_1+r_2x_2) \in im(H)$.
similarly you can prove for $ker(H)$. Try it.