# Annihilator of Noetherian module has Noetherian quotient

## Contents

## 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 be a Noetherian module over a commutative unital ring . Let be the annihilator of . Then the quotient ring is a Noetherian ring.

## Proof

Let be a finite generating set for . Consider a -module map from to given by:

The kernel of this map is precisely , so the quotient is a submodule of .

Since is Noetherian, is Noetherian, and hence is Noetherian (as it is a submodule of a Noetherian module). But being Noetherian as a -module is equivalent to being a Noetherian ring.