Annihilator of Noetherian module has Noetherian quotient

Statement

Verbal statement

Consider a Noetherian module over a commutative unital ring. The quotient of the ring by the annihilator of this module, is a Noetherian ring.

Symbolic statement

Let $M$ be a Noetherian module over a commutative unital ring $R$. Let $I$ be the annihilator of $M$. Then the quotient ring $R/I$ is a Noetherian ring.

Proof

Let $m_1,m_2,\dots ,m_n$ be a finite generating set for $M$. Consider a $R$-module map from $R$ to $M^n$ given by:

$a\mapsto (a{m}_1,a{m}_2,\dots ,a{m}_n)$

The kernel of this map is precisely $I$, so the quotient is a submodule of $M^n$.

Since $M$ is Noetherian, $M^n$ is Noetherian, and hence $R/I$ is Noetherian (as it is a submodule of a Noetherian module). But $R/I$ being Noetherian as a $R$-module is equivalent to $R/I$ being a Noetherian ring.