Von Neumann Regular Ring - Generalizations and Specializations

Generalizations and Specializations

Special types of von Neumann regular rings include unit regular rings and strongly von Neumann regular rings and rank rings.

A ring R is called unit regular if for every a in R, there is a unit u in R such that a=aua. Every semisimple ring ring is unit regular, and unit regular rings are directly finite rings. An ordinary von Neumann regular ring need not be directly finite.

A ring R is called strongly von Neumann regular if for every a in R, there is some x in R with a = aax. The condition is left-right symmetric. 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.