0.1 R-Module and left/right module definitions
Definition 0.1. Consider a ring R with identity. Then a left module ML over R is defined
as a set with two binary operations,
and
such that
-
1.
- (+) + w = +(+w) for all ,,w ∈ ML
-
2.
- +=+ for all ,∈ML
-
3.
- There exists an element 0 ∈ ML such that +0 = for all ∈ML
-
4.
- For any ∈ML, there exists an element ∈ML such that +=0
-
5.
- a ∙ (b ∙) = (a ∙ b) ∙ for all a,b ∈ R and ∈ML
-
6.
- a ∙ (+) = (a ∙) + (a ∙) for all a ∈ R and ,∈ML
-
7.
- (a + b) ∙=(a ∙) + (b ∙) for all a,b ∈ R and ∈ML
A right module MR is analogously defined to ML except for two things that are different in its
definition:
-
1.
- the morphism “∙” goes from MR × R to MR, and
-
2.
- the scalar multiplication operations act on the right of the elements.
Definition 0.2. An R-module generalizes the concept of module to n-objects by employing
Mitchell’s definition of a “ring with n-objects” Rn; thus an R-module is in fact an Rn module
with this notation.
0.2 Remarks
One can define the categories of left- and - right R-modules, whose objects are, respectively, left-
and - right R-modules, and whose arrows are R-module morphisms.
If the ring R is commutative one can prove that the category of left R–modules and the category of
right R–modules are equivalent (in the sense of an equivalence of categories, or categorical
equivalence).
