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Assume R is commutative. Show that an R-module M is irreducible if and only if M is isomorphic (as an R-Module) to R/I where I is a maximal ideal of R. If I remove the hypothesis that R is communtative, is it result still true?

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Yes, this is true, if you replace ideal by right ideal.

See http://en.wikipedia.org/wiki/Irreducible_module#cite_ref-0

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