Difficult# Modules, Simple versus Cyclic : Examples with Group Rings

ABSALG-PHKCDJ

Let $R$ be a **commutative** ring with identity.

For $C=\langle M\rangle$ a cyclic group with a generator $M$, we denote by $R[M]$ the group ring of $C$ over $R$.

In this question, $M$ is the matrix $\begin{pmatrix}0&-1\cr1&\;\;0\end{pmatrix}$ acting on column vectors with 2 entries in the usual way.

Which of the following are true?

Select **ALL** that apply.