Ideal generated by an irreducible element

This article defines a property of an ideal in a commutative unital ring |View other properties of ideals in commutative unital rings

Definition

A proper nonzero ideal in a commutative unital ring is termed an ideal generated by an irreducible element if it satisfies the following equivalent conditions:

• It is the principal ideal generated by an irreducible element.
• It is a principal ideal and every element generating it is an irreducible element.

Facts

• Irreducible element property is not determined by quotient ring: Whether a proper nonzero ideal $I$ in a ring $R$ is generated by an irreducible element cannot be determined by looking at the isomorphism type of the quotient ring $R/I$.