Weibel exercise 1.2.2.: kernels, monics, and monomorphisms are the same in $R$-Mod.

Question

See image below. I just want help proving that all kernels in $R$-Mod are monics.

My attempt:

Let $f : A \to B$ be a map in $R$-Mod. Suppose $i$ is a kernel of $f$, that is: $fi = 0$ and if $fj = 0$ for any other map $j : A'\to A$ then there is a unique map $g: A' \to A$ such that $j = ig$.

Answer 1

Hint: There are two maps fitting into an appropriate diagram for $0: A'\to B$.