This is the first of a series of posts discussing some aspects of modules, particularly over principal ideal domains (PIDs).

Definition

Given a ring $R$, a (left) $R$-module is an abelian group $M$ that has the additonal property that we can define scalar multiplication (on the left) by elements of $R$, such that for $r,s \in R$, $x,y \in M$,

• $r(x+y) = rx + ry$
• $(r+s)x = rx + sx$
• $(rs)x = r(sx)$
• $1_R x = x$.

Here, $1_R$ is the multiplicative identity of $R$.

Modules over fields and PIDs

Many properties of an $R$-module depend on the ring $R$. Generally, the more structure $R$ has, the more restrictions there are on the type of $R$-modules we can have.

If $R$ is a field, then any $R$-module is a vector space over $R$. On the other hand, if $R$ is not even a commutative ring, then

Because of this, it is often said that modules are analogues of vector spaces over arbitrary rings. However, there are many properties of vector spaces that are not shared by arbitrary modules. Most significantly, it doesn’t always make sense to talk about a ``basis’’ of a module.

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