Examples of homomorphisms

Question

Could someone give examples homomorphisms of rings f: R->S and g: S->T such that gof is a monomorphism but g is not?

I tried with the maps from Z (a->na) but can't think of a map such that the composition remains injective.

Answer 1

Consider the composition of rings $\Bbb Z\to \Bbb Z[X]\to \Bbb Z$, the first morphism is the inclusion $f:\Bbb Z\to \Bbb Z[X]$, mapping $n$ into the constant polynomial $n=n\cdot X^0$, the second morphism is $g:\Bbb Z[X]\to \Bbb Z$, the evaluation in zero, $g\to g(0)$. The composition is the identity, thus a monomorphism. But $g$ is of course not injective.

Answer 2

Let $S=\wp(\Bbb N)$, with symmetric difference as addition and intersection as multiplication. Let $R=T=\wp(2\Bbb N)$ with the same operations. Let $f:R\to S:x\mapsto x$, and let $g:S\to T:x\mapsto x\cap 2\Bbb N$. Then $g\circ f$ is just the identity map, but $g$ is clearly not injective.