Irreducible element property is not determined by quotient ring

Statement

In terms of elements

The property of whether a nonzero element $x$ of an integral domain (or more generally, a commutative unital ring) </math>R</math> is an irreducible element cannot be determined by looking at the isomorphism type of the quotient ring $R/(x)$. In other words, we can find integral domains $R,S$ with $x\in R$ irreducible and $y\in S$ not irreducible, such that $R/(x)$ is isomorphic to $S/(y)$.

In terms of ideals

The property of being an ideal generated by an irreducible element is not a quotient-determined property of commutative unital rings: we cannot determine whether an ideal is generated by an irreducible element simply by looking at the isomorphism type of the quotient ring.

Related facts

This fact about irreducible elements stands in contrast to prime elements. An element is prime if and only if the ideal it generates is a prime ideal, which in turn is determined by whether the quotient ring is an integral domain.