Definition 9.2.1 Let R be a commutative ring. Then M is an R module^{2} if it acts just like a vector space except for having the coefficients come from R rather than a field. Thus
Then, just as in the case of vector spaces, 0 in M is unique. So is −m. These assertions only use the fact that M is an Abelian group so there is really no change here. In addition, 0m = 0. This follows from

Now add −0m to both sides. Also
Here we will only consider the case where the ring is a p.i.d. and it will be written as D. Modules will be written in capital letters like A,B,M, etc. Elements of D will usually be written as Greek letters except for primes with will be denoted as p.
Definition 9.2.2 Let the ring D be a p.i.d., a principal ideal domain and let M be a module as discussed above. Then M is called a torsion module if for every m ∈ M, there exists α ∈ D,α≠0, such that αm = 0. The set of all such α is called ann
What is an example of a finitely generated torsion module? The integral domain will be F

In particular, if g

If g

where the degree of r

This observation will be important later. First is an interesting proposition.
Proposition 9.2.3 ann
Proof: It is a subgroup of the additive group of D because if α,β ∈ ann

Also, if α ∈ ann
Note that if the ring is a principal ideal domain, this implies right away that ann
Definition 9.2.4 When M_{i} is a submodule of M we write

when for each i, M_{i} ∩∑ _{j≠i}M_{j} = 0. This is equivalent to the following: If ∑ _{i}m_{i} = 0 for m_{i} ∈ M_{i}, then each m_{i} = 0. This is called a direct sum.