I am reading Lang's Algebra. On page 148:
Let $E$ be a module over $R$. Let $x \in E$. The map $a \mapsto ax$ is a homomorphism of $R$ onto the submodule generated by $x$, and the kernel is an ideal, which is principal, generated by an element $m \in R$. We say that $m$ is a period of $x$. We note that $m$ is determined up to multiplication by a unit (if $m \neq 0$).
(Here $R$ is a principal ideal ring)
I got completely stucked: I cannot come up with any nontrvial examples of the notion of "period". Concretely:
- What does the kernel look like?
- Why is $m$ determined up to multiplication by a unit?
Could anyone give me some examples? (More is better). I would appreciate it a lot.
For instance, what if $R=k[x]$?
The simplest interesting case is $R=\Bbb Z$, for which $R$ modules are just Abelian groups. If an element $x\in{E}$ has finite order $n>0$, i.e., $nx=0$ and no smaller positive integer than $n$ has this property, then the kernel of multiplication by $x$, as map $\Bbb Z\to{E}$, is $n\Bbb Z$, and $n$ (or $-n$) is the period of $x$, according to the given definition.
Another important example is with $R=K[X]$ (with $K$ a field, so that $R$ is a euclidean and therefore principal ideal domain) and $E$ a $K$-vector space, made into a $K[X]$ module by having (multiplication by) $X$ act as some fixed linear operator $T$ on $E$. Then for $v\in{E}$, if the kernel of the map $K[X]\to{E}:P\mapsto P[T](v)$ is non trivial (which happens whenever the subspace generated by all the vectors $T^i(v)$ for $i\in\Bbb N$ is finite dimensional, in particular whenever $E$ is so), it is a principal ideal generated by a single monic polynomial, which one could call the $(T,v)$-minimal polynomial (I'm not sure there is a standard term for it), and in the context of the $K[X]$-module the period of $v$.
By the way it seems the definition assumes that $R$ is a principal ideal ring, for otherwise I see no reason why the ideal would have to be principal.