Answer on Abstract Algebra Question for Hym@n B@ss
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.
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.
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