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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?

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The set of matrices $\mathbb M_{n \times m}$ is a ring when $n = m$. This is because the matrices need to be square in order to be multiplied together.

You are correct that if $F^n$ is taken to be the set of $n \times 1$ column matricies then matrix multiplication gives an action of $\mathbb M_{n \times n}$ on $F^n$ making $F^n$ a module for the ring $\mathbb M_{n \times n}$. This is actually what's called a left module for $\mathbb M_{n \times n}(F)$ because when you multiply a matrix $M$ and a column vector $V$ the matrix is on the left: $MV$.

Here the map looks like $$\mathbb M_{n \times n} \times F^n \to F^n$$ In general if $M$ is a module for the ring $R$ then a left module map looks like $$R \times M \to M$$ There is also a notion of right modules where the ring acts on the module by multiplying on the right. Right module maps look like $$M \times R \to M$$ Note that you can tell which of the two objects is the module by looking at the codomain of the map. The codomain of $\mathbb M_{n \times n} \times F^n \to F^n$ is $F^n$ so matrix multiplication makes $F^n$ the module and $M_{n \times n}$ the ring, you can't switch $F^n$ to be the ring and $M_{n \times n}$ to be the module.

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By definition, the concept of "product" is defined on every ring; this is not true on modules, which apart from being abelian groups, are endowed with an action of the ring on them. In other words, given a module over a ring, you know how to "sum" elements but not how to "multiply" them. A source of confusion can be the fact that every ring is a module over itself: the module structure is canonically induced by the product in the ring: can you show this?

As an example to finish this discussion think of linear spaces, i.e. modules over fields: any linear space $V$ is a module over the ground field $\mathbb K$, while the space of linear endomorphisms on $V$, i.e. $\operatorname{End}_{\mathbb K}(V)$ is a ring.

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