Annihilator of Noetherian module has Noetherian quotient

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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,\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.