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.
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A principal ideal ring is a ring whose left ideals and right ideals each have one generator. For a commutative ring, this says that every ideal is a principal ideal. The examples here use commutative rings. Unlike a principal ideal domain, it may have zero divisors: has this property, with ideals , , and . A finite direct product of rings that are principal ideal rings is again a principal ideal ring. Indeed, the coordinate idempotent elements show that every ideal is a product of coordinate ideals; a tuple of their generators generates the product.