Let $R$ be an integral domain and let $(c)$ be a non-zero maximal ideal in $R$. Prove that $c$ is an irreducible element.
2026-03-25 13:51:38.1774446698
Maximal ideal generated by irreducible element
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Since $(c)$ is nonzero, you know that $R$ is not a field. Suppose $c=ab$; then $(c)\subseteq(a)$ and $(c)\subseteq(b)$. Since $c$ is not invertible, one among $a$ and $b$ is noninvertible, say it's $a$; then $(a)$ is a proper ideal.
By maximality of $(c)$…