So, by definitions, it says that a module is when an abelian group is acted on by a ring. I understand the requirements of a ring, but not what a module is. For example, my teacher gave a module example Mnxn(nxn matrix)XV ->V. I interpret this as saying the matrix is the ring(are all matrices rings?) and the abelian group must be V which he said was an element of F^n. I have a few questions about this example if anyone could help.
What if I flipped this example and my ring was V with matrix as the abelian group, this would be incorrect since matrices are not abelian right? Does a module only work when the right side is abelian and the left a ring? Does this imply that V in my example is abelian? Is it also correct to say a module is an action map while a ring is a set with certain properties?