Answer to Question #17098 in Abstract Algebra for Hym@n B@ss

Question

Show that, if M is a simple module over a ring R, then as an abelian group, M is isomorphic to a direct sum of copies of Q, or a direct sum of copies of Zp for some prime p.

Answer 1

By Schur’s Lemma, the R-endomorphism ring of M is a division ring D. Let F be the prime field of D. We may view M as a D-vector space, so M is also an F-vector space. As such, M is isomorphic to a direct sum of copies of F. This gives the desired conclusion since we have either F∼ Q, or F ∼ Zp for some prime p.