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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.

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    yes of course... it came to my mind when i was not on the computer... however.. thank you... if you post it as an answer i will accept it immediately.2012-01-19

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This CW post intends to remove the question from the unanswered queue.


As Thomas Andrews noted in the comments, there is a (trivial) counterexample, taking the inclusion $f:A\to A\oplus D$ and the projection $g:A\oplus D\to A$. Here their composition $gf$ is the identity, hence essential, but $f$ is not essential and $g$ is not a monomorphism.