Is the quotient of a finitely generated module finitely generated?

Question

Is the quotient of a finitely generated module finitely generated ?

I have difficulties to find a counterexample... or to prove that it is finitely generated.

Answer 1

Suppose $M$ is a finitely generated module over $A$, with generators $x_1,\ldots, x_n$. Then, consider the quotient $\pi: M\to M/N$, for $N$ a submodule of $M$. Let $\tilde{x}_i=\pi(x_i)=x_i+N$. Then $\tilde{x}_1,\ldots, \tilde{x}_n$ generate $M/N$.

To see this, let $m+N$ be given in $M/N$ with $m\in M$. $$m=\sum_{i=1}^n a_ix_i$$ for some $a_i\in A$. So, we have that $$ m+N=\pi(m)=\sum_{i=1}^n a_i\tilde{x}_i.$$

Answer 2

If a module $M$ is generated by $\{m_1,\ldots,m_k\}$ and if $N$ is a submodule, then $M/N$ is generated by $\{m_1+N,\ldots,m_k+N\}$.