Definition. Let be rings. For every we let be the natural projection. Then is called a subdirect product of if the following conditions are satisfied:
1) There exists an injective ring homomorphism
2) For every the map is surjective.
Note. Suppose that are some ideals of and put Then we can define by Clearly the second condition in the above definition is satisfied. Thus is a subdirect product of if and only if is injective, i.e.
Remark 6. If is a minimal prime ideal of the ring then is multiplicatively closed iff , for all
Proof. Suppose that for any and Let be the set of all elements of which are a finite product of some elements of Clearly is multiplicatively closed, and is multiplicatively closed iff So we’ll be done if we show that . Let We have because Therefore, by Zorn’s lemma, has a maximal element and is a prime ideal of Since we have and thus Thus because is a minimal prime. So , which means Hence
Remark 7. If is reduced and is a minimal prime, then is a domain.
Proof. Clearly is a domain iff is multiplicatively closed. Let be as defined in Remark 6. By that remark, we only need to show that So suppose that for some , where the integer is assumed to be minimal. Then by, Remark 1, we have Now, since is prime, cannot be a subset of because otherwise we’d have either or which is clearly nonsense. Thus Let Then
Hence which contradicts the minimality of
The Structure Theorem For Reduced Rings. A ring is reduced iff is a subdirect product of domains.
Proof. If is reduced, then, by Remarks 5 and 7, is a subdirect product of the domains where is the set of all minimal prime ideals of Conversely, suppose that is a subdirect product of domains and is an injective ring homomorphism. Suppose that and Let Then Thus for all and so for all because every is a domain. Hence and so is reduced.