Noetherian Rings and Modules

More precisely, an *-module* is an additive abelian
group equipped with a map
such that for all
and all
we have
,
,
, and . A *submodule*
is a subgroup of that is preserved by the action of .
An *ideal* in a ring is an -submodule
,
where we view as a module over itself.

A *homomorphism* of -modules
is a abelian group
homomorphism such that for any and we have
. A *short exact sequence* of -modules

Notice that any submodule of a noetherian module is also noetherian. Indeed, if every submodule of is finitely generated then so is every submodule of , since submodules of are also submodules of .

- is noetherian,
- satisfies the ascending chain condition, and
- Every nonempty set of submodules of contains at least one maximal element.

**
:** Suppose 3 were false, so there exists
a nonempty set of submodules of that does not contain a
maximal element. We will use to construct an infinite ascending
chain of submodules of that does not stabilize. Note that is
infinite, otherwise it would contain a maximal element. Let be
any element of . Then there is an in that contains
, otherwise would contain the maximal element .
Continuing inductively in this way we find an in that
properly contains , etc., and we produce an infinite ascending
chain of submodules of , which contradicts the ascending chain
condition.

**
:** Suppose 1 is false, so there is a
submodule of that is not finitely generated. We will show
that the set of all finitely generated submodules of does not
have a maximal element, which will be a contradiction. Suppose
does have a maximal element . Since is finitely generated and
, and is not finitely generated, there is an such that
. Then is an element of that
strictly contains the presumed maximal element , a contradiction.

Next assume nothing about , but suppose that both and are noetherian. If is a submodule of , then is isomorphic to a submodule of the noetherian module , so is generated by finitely many elements . The quotient is isomorphic (via ) to a submodule of the noetherian module , so is generated by finitely many elements . For each , let be a lift of to , modulo . Then the elements generate , for if , then there is some element such that is an -linear combination of the , and is an -linear combination of the .

The rings and are isomorphic, so it suffices to prove that if is noetherian then is also noetherian. (Our proof follows [Art91, §12.5].) Thus suppose is an ideal of and that is noetherian. We will show that is finitely generated.

Let be the set of leading coefficients of polynomials in . (The leading coefficient of a polynomial is the coefficient of highest degree, or 0 if the polynomial is 0; thus has leading coefficient .) We will first show that is an ideal of . Suppose are nonzero with . Then there are polynomials and in with leading coefficients and . If , then is the leading coefficient of , so . Suppose and with . Then is the leading coefficient of , so . Thus is an ideal in .

Since is noetherian and is an ideal, there exist nonzero that generate as an ideal. Since is the set of leading coefficients of elements of , and the are in , we can choose for each an element with leading coefficient . By multipying the by some power of , we may assume that the all have the same degree .

Let be the set of elements of that have degree strictly less than . This set is closed under addition and under multiplication by elements of , so is a module over . The module is the submodule of the -module of polynomials of degree less than , which is noetherian because it is generated by . Thus is finitely generated, and we may choose generators for .

We finish by proving using induction on the degree that every is an -linear combination of . If has degree 0, then , since , so is a linear combination of . Next suppose has degree , and that we have proven the statement for all elements of of degree . If , then , so is in the -ideal generated by . Next suppose that . Then the leading coefficient of lies in the ideal of leading coefficients of elements of , so there exist such that . Since has leading coefficient , the difference has degree less than the degree of . By induction is an linear combination of , so is also an linear combination of . Since each and lies in , it follows that is generated by , so is finitely generated, as required.

Properties of noetherian rings and modules will be crucial in the rest of this course. We have proved above that noetherian rings have many desirable properties.