The analogue in ring theory for the idea of a group action is that of a module. Again we want every element of the ring to behave like a function on a set and for multiplication to correspond to composition of functions, but now we have the addition floating around and we’d like to include it in the structure as well. We’ll handle this by letting the ring act on an abelian group.
So let’s take a ring and an abelian group . We say that has the structure of a left -module if each element acts as a linear function from to itself. We write for the effect of this function on an element . Linearity here means that . We also require that the ring structure play its part
- if the ring has an identity element
We can say all this another way. Since this action of on is linear in both the ring and the abelian group we get a linear function from the tensor product just like we had one for the multiplication of the ring: . Not every such function will work, though. We also need that the following diagram commutes:
Around the top of the diagram we use the module action twice, while around the bottom we first multiply in the ring and then use the action once. If the ring has an identity we also require that the following diagram commute
where on the top we use the unique linea function sending the integer to the identity in , and the diagonal is the canonical isomorphism from to .
And there’s another way to say it. Remember that every abelian group comes equipped with the endomorphism ring . An -module structure on is a ring homomorphism .
All three ways of defining a module — from the raw axioms, as a linear function , or as a homomorphism — are useful in various situations, and it’s important to be able to slide back and forth between the different pictures.
Of course, the fact that I said left -module above should immediately lead you to think about right -modules. These are the same, except that the module action is written and satisfies — the order in which the factors are applied is reversed. Over a commutative ring we often ignore this distinction, since we can always switch the order of any product.
Every ring immediately has the structure of a left and a right -module, using multiplication as the action. This is most clearly seen from the tensor-product definition, where the commuting square just expresses the associativity of ring multiplication. On the other hand, every abelian group immediately has the structure of an -module. This is trivial in the homomorphism picture: just pick the identity homomorphism.
Somewhat more importantly, every abelian group is naturally a -module. We define — adding up copies of the group element . Everything we say for modules in general will apply to abelian groups, and in fact much of what we’ve already said about abelian groups will extend to other modules.
At this point I want to make a point about language. The terms “abelian group” and “-module” are interchangeable, but there are cases in which I feel the context clearly indicates using one or the other. Later we’ll come to (and many reader have already seen) “homology groups”, which are more naturally modules than groups. Similarly, the one-dimensional circle, torus, and real projective space are all semantically very different, even though they happen to be equivalent in many situations. I think that this imprecision can lead to confusion on the part of a student, so I’ll try my best to use the most apropriate of equivalent terms.