nLab principal ideal ring

Idea

A version of a principal ideal domain where the commutative ring is not required to be an integral domain.

Definition

A commutative ring $R$ is a principal ideal ring if every ideal in $R$ is a principal ideal, or equivalently that the set of ideals is isomorphic to the quotient multiplicative monoid $R / R^\times$.

Examples

• Every principal ideal domain is a principal ideal ring.

• Given a prime number $p$ and natural number $n \gt 1$, the prime power local ring $\mathrm{Z} / (p^n)$ is a principal ideal ring where the zero divisors form a principal ideal generated by $p$.

See also

• principal ideal domain