Math 6310 • Algebra

Let \(R\) be a ring and let \(M\) be an irreducible \(R\)-module. Prove that one of the following
statements is true about \(M\) considered as an abelian group:
\begin{enumerate}
\item[(1)] \(M\) is the direct sum of (perhaps infinitely many) copies of the additive group of all
rational numbers.
\item[(2)] \(M\) is the direct sum of (perhaps infinitely many) copies of a cyclic group of order
\(p\) for some prime \(p\).
\end{enumerate}