# Relaxing the definition of a von Neumann regular ring

Hereinafter, all rings are assumed to be unital but not necessarily commutative. A well-known class of rings are von Neumann regular rings, that is, rings $R$ such that for each $a\in R$ there is an $x\in R$ satisfying $a=axa$. They include many classical rings, but the motivation for the definition comes from ...analysis.

It is not very hard to spot that a ring is von Neumann regular if and only if each its principal left ideal is generated by an idempotent element. I am interested whether anyone has studied rings for which each maximal left ideal, which is principal, is generated by an idempotent.

Probably you'll ask for examples of rings with this property which are not von Neumann regular: the easiest one is the algebra $C(K)$ of scalar-valued continuous functions on an infinite compact space.

I am interested also in rings which are not local but every maximal left ideal is principal.

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