Ideal generated by an irreducible element

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 fact about::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 $$R/I$$.