Structure theorem for commutative Noetherian rings

Question

We have Structure theorem for commutative Artin rings which is as follows :

An Artinian ring $A$ is uniquely up to isomorphism a finite direct product of Artin local rings.

I would like to know if we have some similar theorem in case of commutative Noetherian rings. If not in this generality, atleast in some special case which is stronger than being commutative Artinian ring.

Any reference is most welcome.

I know that any Noetherian reduced ring is embedded in a finite product of fields.

As $A$ is Noetherian, there are finitely many minimal prime ideals. Intersection over all minimal prime ideals is same as that of intersection over all prime ideals which is nil radical, in case of reduced ring is zero.

So, we have an injective map $A\rightarrow A/\mathfrak{p}_1\times A/\mathfrak{p}_2\times\cdots\times A/\mathfrak{p}_n$ which then can be embedded into field of fractions of integral domains $A/\mathfrak{p}_i$ and their products.

I am looking for some theorem stronger than this.

Answer 1

No, there is no such nice theorem for all commutative Noetherian rings.

You can find special results, though, for commutative and noncommutative Noetherian rings.

I think the best-studied class of Noetherian rings in terms of structure are hereditary Noetherian rings, especially hereditary Noetherian prime rings a.k.a. Dedekind prime rings. (e.g.this For material on this I would consult texts by Robson on the subject, as well as Chatters & Hajaranvis' book on Rings with Chain conditions.

I also think the structure of Noetherian serial rings is well understood, but I can't tell you how much, if at all, this overlaps with the study of hereditary Noetherian rings.

You can also discover other special cases like this one.

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The theorem on commutative reduced Noetherian rings that you referenced is a special case of "vague" structure theorems available for all rings.

The lower nilradical of a ring, given by the intersection of all prime ideals (or all minimal prime ideals)) is denoted $Nil_\ast (R)$. You can say that $R/Nil_\ast (R)$ is a subdirect product of prime rings.

Similarly, if $J(R)$ denotes the Jacobson radical, you can say that $R/J(R)$ is always a subdirect product of right primitive rings.

Different conditions may help limit the number of factors in the subdirect sum, or actually allow you to conclude that the canonical map is onto the image of the product.