Two Examples of Cyclic Modules

Question

Give an example of:

(a) a cyclic $\mathbb R[x]$ module that is three-dimensional over $\mathbb R$.

(b) a cyclic $\mathbb Z[x]$ module with a submodule that is not cyclic.

I've been studying for a test coming up, and I've been having trouble coming up with examples like these. Any help is greatly appreciated.

Answer 1

Question (a) Let $M=\mathbb{R}^3$, and for each $(a,b,c)\in M$, let $x\cdot(a,b,c)=(0,a,b)$. We can extend this definition to get a map $\mathbb{R}[x]\times M\to M$ that will make $M$ into an $\mathbb{R}[x]$ module. The module will be cyclic, since it will be generated by $(1,0,0)$.

Question (b) Let $M=\mathbb{Z}[x]$. $M$ is a cyclic $\mathbb{Z}[x]$ module. Let $I=(2,x)$, i.e. $I$ is the ideal of $\mathbb{Z}[x]$ that is generated by $2$ and $x$. Then $I$ is a submodule of $M$, but it is not cyclic.

Answer 2

Hint: quotient rings of the base ring are very common examples of cyclic modules, and ideals of these rings are very common examples of submodules.