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Abstract Algebra

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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.

A

Every cyclic $R$-module is simple.

B

Every simple $R$-module is cyclic.

C

The left $\mathbb{Z}[M]$-module $\mathbb{Z}^2$ is simple.

D

The left $\mathbb{Q}[M]$-module $\mathbb{Q}^2$ is simple.