Ideals and quotients
Now that we know what a homomorphism of rings is, we can do the same thing we did for groups: form quotients. Remember that a ring is already an abelian group if we just consider the additive structure. We’re going to use that and see how the multiplication restricts us.
A subring of a ring is (as you might guess) a subset of the ring that’s still a ring, using the same operations. It’s got the sum of any two of its elements, the products, the negatives… In particular it has to be a subgroup of the additive group structure. Even better, since this is an abelian group any subgroup is normal, so we don’t have to worry about that holding us back.
Since (forgetting all but the additive structure) the ring is a group and the subring is a subgroup, we can slice into cosets of . I’ll write a general one as , since the group is additive. Recall that this is the collection of elements of we get by adding to every element of . Now, since is a normal subgroup of (additive structure only) there’s clearly an abelian group structure on the set of cosets. This is just how we made the quotient group before. But we also want a multiplication on the set of cosets.
Well, the obvious way to multiply cosets and is to use . But as usual we have to be careful that our choice of representative doesn’t throw us off. What if we chose and instead of and , where and are in ? Then we get the coset . If this is going to be the same as , we’d better have a few things cancel.
For one thing, pick , which is in every subring. Now to have we have to require that land back in for any in the subring and any at all in the ring . That is, has to not only contain all the products of its elements, but their products with any other elements of . If we pick we can see that also has to be in , so we have to have those products as well to make multiplication of cosets make sense.
This special kind of subring we call an “ideal”. To restate: an ideal of a ring is a subgroup of the additive structure of which contains all products on either side with any element of . Some authors further distinguish “left ideals” and “right ideals”, which contain products with an arbitrary element of on the left side or the right side, respectively. Our ideal is then both a left and a right ideal. Of course, for a commutative ring all ideals are two-sided like this.
At first glance it rather seems like an ideal has to contain an awful lot of the ring, and often it does. Still, enough is left over to be interesting, as we’ll see.