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Proposition: Principal Ideals being Maximal Ideals
Let $(R, + ,\cdot)$ be an integral domain, $a\in R$ with $a\neq 0.$ A principal ideal $(a)\lhd R$ is a maximal ideal among all principal ideals, if and only if $a$ is irreducible in $R.$
This holds especially, if $R$ is a principal ideal ring.
Table of Contents
- Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013