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.