Annihilator of Noetherian module has Noetherian quotient

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

$$a \mapsto (am_1,am_2,\ldots,am_n)$$

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.