Let \(R\) be a ring and let \(M\) be an \(R\)-module. \(M\) is decomposable if there exist nonzero \(R\)-modules \(M_1, M_2\) with \(M \cong M_1 \oplus M_2\). \(M\) is indecomposable if \(M\neq 0\) and \(M\) is not decomposable: whenever \(M \cong M_1 \oplus M_2\), either \(M_1 = 0\) or \(M_2 = 0\).
Decomposable and indecomposable modules
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