Annihilator of Noetherian module has Noetherian quotient
From Commalg
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.