First off, a submodule of a left -module is just an abelian subgroup that’s closed under the action of . That is, for any and , we have . A submodule comes with an inclusion homomorphism .
Now if we take an -module and a submodule we can just consider and as abelian groups and form the quotient group. Remember that every subgroup of an abelian group is normal, so the quotient is again an abelian group made up of the “slices” . It turns out that this is again an -module. Just use the action . If we chose a different representative of the slice, then we’d get , which represents the same image, so this action is well-defined. Quotient modules come with projection homomorphisms .
Actually there’s one sort of submodule we’ve already seen. Remember that every ring is a left module over itself. Then the left submodules of are exactly the left ideals of ! If we have a two-sided ideal then we get a left action of on the quotient module and a right action as well, since is also a right submodule. Then we can take the tensor product and get a linear function from multiplying representatives. Presto! Quotient ring!
We can use ideals to give submodules and quotient modules of other -modules too. Take a ring with left module and a left ideal . Then we can restrict the action of on to the ideal to get . This is clearly an abelian subgroup of , and it turns out to be a submodule too. Indeed, we see that . Then we can make the quotient -module as above. Even better, if is two-sided this is actually a module over : use the right -module structure on and the left -module structure on and tensor to get . Then we can get a linear function by choosing representatives and showing that the choice is immaterial.
I promised an isomorphism theorem. Well, I’ll state it, but the proof is pretty much exactly the same as the two we’ve seen before so I’ll leave you to review those. Any homomorphism of left (right) -modules factors as the composition , where the first arrow is the projection homomorphism, the third is the inclusion homomorphism, and the middle arrow is an isomorphism. We call the submodule the kernel of the homomorphism and the submodule the image of the homomorphism. Notice that there’s no restriction on the sorts of submodules that can be kernels here. For groups a kernel is a normal subgroup, and for rings a kernel is a two-sided ideal, but any submodule can be the kernel of a module homomorphism. This leads to a few more definitions that come in handy. The quotient module is called the coimage, and the quotient module is called the cokernel. Thus we see that the coimage and the image of any homomorphism of modules are isomorphic.