= Principal ideal ring
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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>: $\mathbb Z/4\mathbb Z$ has this property, with <ideals> $(0)$, $(2)$, and $(1)$. 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.
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