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.
 A: I haven't seen and couldn't find anything on the first type of ring you described. However, generalizations of VNR rings are plentiful, so that is not saying much. 
You might look at stuff on cyclically presented simple modules/ideals (or even finitely presented ones). The equivalent condition that I was thinking of (which is admittedly less nice and more complicated than yours) for your condition is "All simple cyclically presented modules are projective."
For the second type of ring you describe, you will have to stay away from (left and right) Noetherian rings, because it is known that such rings are principal right ideal rings. I imagine you are probably already aware of the commutative version (Kaplansky's theorem). Lam and Reyes generalized it to noncommutative Noetherian rings in one of my favorite papers here.
If you are working with rings of continuous functions over noncompact spaces then you are nowhere near Noetherian rings :) At times like this I wish I owned a copy of that book by Gillman and Jerison...
