Linear Algebra over a Ring/Homomorphism and dual modules

Proposition (multiple of module homomorphism by a ring element over a commutative ring is module homomorphism):

Let be modules over the commutative ring . Let be a homomorphism of -modules, and let . Then the function

is an -module morphism too.

Proof: Let and . Then and since is commutative, also .

{{definition|homomorphism module|Let be a commutative ring

Definition (dual):

Let be a commutative ring, and let be an -module. Then has a module structure given by addition and pointwise multipliction (since the category of modules over a ring is additive and the addition that is implied is compatible with pointwise multiplication, under which is closed), and this module, denoted more briefly by , is called the dual of .

Proposition (the category of modules is abelian):

Let be a ring. Then the category of left -modules is abelian. Also, the category of right -modules is abelian.

Proof: Existence of kernels and cokernels has been dealt with, and the same holds for existence of binary biproducts, since then product and sum coincide in this category. Also, the Noether isomorphism theorem holds. Then, the category is additive. Indeed, the summation on the homomorphism group indicated above is precisely the addition in the categorical sense.