User:Nawajkc/sandbox

=== The canonical reference is to the notes by Alexandra Seceleanu.

Conventions and Notations:

===
 * 1)  Every ring is unital, possibly non-commutative, and every ring-map takes $$ 1 \to 1 $$.
 * 2)  Unlike in the notes, the Categories will be denoted by $$ \mathcal{C} $$. To avoid confusions, let the categories of Sets, Abelian Groups, Groups, Rings, and R-modules be denoted by $$ \mathcal{S}, \mathcal{Ab}, \mathcal{G}, \mathcal{R},\mathcal{_RMod} $$ respectively.

What is a left R-module M? It is simply the pairing $$ \dot : R \times M \to M $$ such that this pairing has the nice, linear properties that every undergraduate knows. However, a more subtle point of view would be to say, that a R-module M is simply a ring homomorphism $$ R \to End(M) $$. First of all, $$ End(M) $$ is obviously a ring of homomorphisms from $$ M \to M $$ as the underlying group is Abelian.