Submodules and Homomorphisms

Just as a subgroup is a group within a group or a subfield is a field within a field, a submodule is a module within a module.  That is, if is an -module, we say that is a submodule of if is a subgroup of and is closed under multiplication by elements of .  This can be summarized by saying that for every and every , the following two properties hold.

1. 
2. 

In the case of groups, we could only form a quotient group if the subgroup we were modding out by as sufficiently “nice” (i.e., was a normal subgroup); likewise, in the case of rings we required that a subring be nice (an ideal) in order to form a quotient ring.  With modules however, we can always form the quotient module with a submodule.  We can do this since, as is an abelian group under addition, all subgroups are normal and we can form the quotient group .  This is naturally an abelian group, so in order to turn this into an -module we have to define multiplication of elements in and , which we do in the most obvious way: .  To check that this is in fact an -module, we simply verify that the three axioms of a module hold.

1. 
2. 
3. 

Finally, if is a unitary module, then so too is : .

We say a map between two -modules is a homomorphism if is a group homomorphism with the additional property that for each , .  As you would expect, the kernel of this homomorphism is a submodule of , and the image is a submodule of .

Supposing that , we define to be the smallest submodule of containing , which is naturally the intersection of all submodules containing

If is a finite set, we may write in place of , and in the event that is a singleton, we say that is a cyclic submodule of .  We call the elements of the generating set of the submodule, and call the elements of the generators of this submodule.