TITLE: Weibel exercise 1.2.2.: kernels, monics, and monomorphisms are the same in $R$-Mod.
QUESTION [6 upvotes]: 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 $j = ig$.

REPLY [2 votes]: Hint: There are two maps fitting into an appropriate diagram for $0: A'\to B$.