Our goal here is to prove that the quotient of a graded ring by a graded ideal is graded. We will approach this from a slightly more general perspective.
Let be a commutative ring with 1 and let be a semigroup. Suppose we have a family of -bimodules such that for all and . Suppose further that we have a family of -bilinear mappings such that for all and , , and . Finally, define an operator on the -module by .
We claim that is a ring under this multiplication. It suffices to show that is associative and distributes over addition. To that end, let .
- Note the following.
= = = = = = = .
So is associative.
- Note the following.
= = = = = .
So distributes over from the left; distributivity from the right is proved similarly.
So is a ring under this multiplication; we will call this the -graded sum of the , and denote it by .
Now we claim that if is a monoid with identity , and if , and finally if and for all , , and , then in fact is a unital ring and moreover an -algebra via the injection map . To see this, consider the element with and otherwise. Certainly then for all . So is a unital ring. Moreover, note that since for all , we have . In particular, this ring is an -algebra.
Next we prove that this algebra has an appropriate universal property. Suppose is an -algebra and that we have a family of -module homomorphisms such that for all and all and . Then there exists a unique -algebra homomorphism such that for each . It suffices to show that the unique induced module homomorphism (via the universal property of coproducts of modules) is an -algebra homomorphism. To that end, let . We have for each . Then . (Recall that .)
Now suppose we have a family of submodules , with , such that for all . Certainly is an ideal; we will call ideals of this form -graded.
Let , , and be as above. For each pair , define by . We claim that is well-defined and -bilinear. To see well-definedness, note that if and , then by bilinearity we have and . Thus , and so is well-defined. Bilinearity is clear. Moreover, it is straightforward to show that . Thus we may construct the -graded sum .
Let , and for each pair , let denote the restriction of to . Let denote the set of these restrictions. Then the ideal is in fact .
We claim that . To see this, it suffices to show that the usual module isomorphism given by preserves multiplication. To that end, note that , as desired.