Definition Let $f$ be a right $R$ module monomorphism. Say $f:A\to B$. We say that $f$ is essential if its image $f(A)$ is essential in $B$, that is, for any $C\leq B$ non zero, $f(A)\cap C\neq 0$.
Continuing in my studying injective modules and injective envelops I've encountered (and proved) the following fact, namely that if $f,g$ are two right $R$ module monomorphism, say $f:N\to M$ and $g:M\to P$, then $gf$ is an essential monomorphism if and only if both $f$ and $g$ are essential monomorphism.
Question Assume now $f,g$ are just right $R$ module homomorphism. Could it be possible for $gf$ to be still an essential monomorphism, without both $f$ and $g$ be essential monomorphism?
I think this should be true, but i cannot find an example.