Let $R$ be a commutative ring with unity which is not a principal ideal ring . Then is it true that $\exists 0\ne x,y \in R$ such that the ideal $\langle x, y \rangle$ is not a principal ideal ?
2026-03-30 12:22:34.1774873354
A commutative ring with unity which is not a PIR has a non-trivial ideal generated by two elements which is not a principal ideal?
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No it is not true. There exist rings which have the property finitely generated ideals are all principal. These are called Bezout rings.
There are such rings that aren't principal ideal rings, and any of those would be a counterexample to this proposition.
A specific example would be a nonprincipal von Neumann regular ring, like an infinite product of fields.