M is finitely generated but $\ker(f)\subset M$ is not.

Question

So I'm searching for an homomorphism of $A$-modules $f:M\mapsto N $ and some A-modules $M$, $N$ such both are finitely generated but $\ker f\subset M$ is not.

Thanks.

Answer 1

Consider $A=k[x_1,x_2,....]\longrightarrow k$ sending $x_i\mapsto 0$. These are modules over $A$, finitely generated. Kernel is $<x_1,x_2,\cdots>$.