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

Question #17098

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.

Expert's answer

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**∼*Z*p*for some prime*p*.
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