Module Theory, Part I: Introduction, Module Homomorphisms

September 14th, 2017

Definition: Let R be a ring. A module M over a ring R is an abelian group (with operation +) and a map R×M→M which satisfies distributivity and associativity. If R has a 1, then we require that 1m=m for each m∈M.

These axioms should look fairly familiar. if R is a field, then a module is exactly a vector space over R. A module is nothing more than a generalization of vector spaces.

Definition: A submodule is a closed subgroup N of an R-module M which is closed under the action of R.

Example

A significant example is modules over Z. The action of an integer on m∈M is defined straightforwardly as:

nm=m+m+⋯+m

Where we are adding m to itself n times. This is the only possible action of Z over M, because of associativity and distributivity. What we have from this is that:

Z-modules are exactly abelian groups.

In particular, Z-submodules are exactly submodules of subgroups.

Example

By associativity, we can define a module over F[x], where F is a field. We simply need to define how 1,x act on elements in the module. Let V be a vector space over F – we will make V an F[x] module, by identifying the action of x with a linear transformation T:V→V.

Conversely, if we have any module V over F[x], then in particular V is a module over F. But we know that:

x(v+w)=xv+xwx(av)=ax(v)

So this means that indeed x is a linear transformation. So there is a natural isomorphism between vector spaces V over F equipped with a linear transformation T and modules V over F[x].

Consequently, the F[x]-submodules of V are exactly vector subspaces of V which are invariant under T.

Proposition

A nonempty N of an R-module M is a submodule iff x+ry∈N for each x,y∈N, r∈R.

Let r=−1 and we get the subgroup criterion. Let x=0 and we get closure under elements of R. The converse case is fairly straightforward.

Quotient Modules and Module Homomorphisms

Definition: Let R be a ring and M,N are R-modules. Then an R-module homomorphism φ is a map from M to N so that:

φ(x+y)=φ(x)+φ(y)φ(rx)=rφ(x)

As expected, an isomorphism is surjective as well as injective. The kernel and images are respectively submodules of M,N as expected. Finally, we define HomR(M,N) to be the set of all R-module homomorphisms from M to N.

For example, Z-module homomorphisms are simply abelian group homomorphisms (since the second criterion is implied by the first above). Over a field, the F-module homomorphisms are simply linear transformations between vector spaces. Note, however, that R-module homomorphisms where R is a ring do not necessarily have any connection to ring homomorphisms – specifically because there is no requirement that a module homomorphism send identity to identity.

Proposition

HomR(M,N) is an R-module.

We can define addition and multiplication in the usual way:

(φ+ψ)(m)=φ(m)+ψ(m)(rφ)(m)=r(φ(m))

Furthermore, if M=N then we can have a well-defined ring structure; multiplication is just composition. Indeed, HomR(M,M) is a ring with identity – and indeed it has a special name.

Definition: The ring HomR(M,M) is called the endomorphism ring of M and is denoted End(M) or EndR(M).

Proposition

Let R be a ring and let M,N be R-modules with N a submodule of M. Then M/N (an abelian quotient group) can be made into a module over R by defining:

r(x+N)=rx+N

And we have a natural projection map π:M→M/N with kernel N.

Finally, we define the sum of two modules:

A+B={a+b : a∈A,b∈B}

So that we can once more define the isomorphism theorems.

Theorem (Isomorphism Theorems)
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