Algebras and why modules are a thing

A $k$ -algebra $A$ where $k$ is a field, is a ring with the added structure of being a vector space over $k$ (with respect to the same addition) where scalar multiplication behaves nicely with the ring multiplication. $\mu(ab) = (\mu a)b = a(\mu b)$ for $\mu \in k$ and $a, b \in A$ . The atypical example is a matrix algebra of dimension $n$ with matrix elements in $k$ denoted $M_{n}(k)$ . There is a notion of algebra morphism which is not too hard to figure out with the above or a quick google, so we have a notion of isomorphism.

Matrix algebras are very special, they have vectors in which they can act on. For an $n$ -dimensional matrix algebra these objects are the tuples $(x_1, \cdots, x_n)$ where $x_i \in k$ . As a collection, they are denoted $k^n$ . These form a vector space and as algebras $M_{n}(k) \cong \text{Aut}(k^n)$ . Where the algebra operation in $M_n(k)$ is matrix multiplication and $\text{Aut}(k^n)$ is the algebra of vector space automorphisms $k^n \longrightarrow k^n$ with the algebra operation being composition.

The definition of an $A$ -module can be found elsewhere but to motivate its origins, it is precisely the articulation of assigning some ‘vectors’ in some vector space $V$ to a general $k$ -algebra $A$ which $A$ can act on $V$ similarly to matrices on tuples. If you struggle to remember the definition of an $A$ -module just think like this and you should be able to reconstruct it.

Modules can even be defined for a ring $R$ , with the relaxation of the structure of a vector space. And so we can assign ‘vectors’ to rings by looking at $R$ -modules. So we can ask the question, what are $\mathbb{Z}$ -modules? These are in fact abelian groups (why?) and so the collection of possible ‘vectors’ for $\mathbb{Z}$ can be thought of as just abelian groups.

I think that’s pretty nice.

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