Principal ideal ring

A **principal ideal ring** (PIR) is a type of ring in which every ideal is a principal ideal. This means that for any ideal \( I \) in the ring \( R \), there exists an element \( r \in R \) such that \( I = (r) = \{ r \cdot a : a \in R \} \). In other words, each ideal can be generated by a single element.