Von Neumann regular ring
= Von Neumann regular ring
{c}
A ring is von Neumann regular if every element $a$ has an element $b$ with $a=aba$. In a <commutative ring>, this is $a=a^2b$. An arbitrary <direct product of rings> whose factors are <fields> has this property by taking inverses in each nonzero coordinate. Every <localization at a prime ideal> of a commutative von Neumann regular ring is a <field>.