Strongly von Neumann regular rings are unit regular. Every strongly von Neumann regular ring is a subdirect product of division rings. In some sense, this more closely mimics the properties of commutative von Neumann regular rings, which are subdirect products of fields. Of course for commutative rings, von Neumann regular and strongly von Neumann regular are equivalent. In general, the following are equivalent for a ring R: R is strongly von Neumann regular R is von Neumann regular and reduced R is von Neumann regular and every idempotent in R is central Every principal left ideal of R is generated by a central idempotent Generalizations of von Neumann regular rings include π-regular rings, left/right semihereditary rings, left/right nonsingular rings and semiprimitive rings. See also Regular semigroup Weak inverse Notes References Further reading Ring theory John von Neumann