In this post I will talk about how to compute the product and tensor product of quotient rings and . This sort of thing is usually left as an exercise (especially the first Corollary) and not proved in full generality in algebra courses, although it is not hard.
In all that follows is a commutative ring with identity and and are ideals of .
Lemma: If , there is a natural map .
Proposition: The natural map is an isomorphism of -algebras.
Proof: To see surjectivity, notice that generates as an -module, and , since the map is a ring homomorphism. To see injectivity, notice that every element
is equal to the pure tensor where . If , then . So , . Then .
Proposition (Chinese Remainder Theorem): The natural map is injective. If , it is also surjective, and thus an isomorphism.
Proof: To see injectivity, notice that if , then and , so so . To see surjectivity, note that implies there exist , , such that , for any . Consider , and set and . Then , so .
Corollary: If , . In particular, by applying this repeatedly we have .
Also, note the following fact:
Proposition: If , then .
Proof: Clearly . To go the other way, note that means that there exist , , and such that . So, consider an element . Then we have . Since , , and since , . So .
Remark: One can also think of this in terms of Tor: , and when this Tor group vanishes.