In von Neumann regular rings every element has a von Neumann inverse. Are there many rings in which these inverses are unique for non-zero elements? Are there any such domains that are not skew fields? Are there any such non-domains?
Oh, you're right, sorry. I want to assume that. I would like to know if
1) there are any von Neumann regular domains which are not skew fields, and whose non-zero elements have unique von Neumann inverses;
2) there are any von Neumann regular rings which are not domains, and whose non-zero elements have unique von Neumann inverses.
ok, i can't believe that i thought the answer was yes! the answer is actually no. to see this, let be a ring with this property that for every nonzero element the equation has a unique solution in then will have no non-trivial idempotent element because if where then the equation will have (at least) two solutions and so now let and let be such that then and because otherwise which is false. thus is a nonzero idempotent of and so hence is a division ring.
I think I must have used "von Neumann inverse" incorrectly. I really should be more careful when I ask questions here. I'm very sorry.
A von Neumann inverse of as I understand it, is an such that
So a unique inverse would be the unique solution of the system of equations. The first equation alone could have more solutions.
Maybe I will tell you why I'm asking this question. In semigroup theory there are regular semigroups and inverse semigroups. Regular semigroups are those which satisfy the von Neumann regularity condition. (It's well-defined for semigroups, since the condition doesn't employ addition.) Inverse semigroups are those in which every element has a unique weak inverse (in the sense defined above -- the two conditions). It's a very rich theory (of which I know barely anything), and I was wondering if it had any bearing on ring theory.
i just realized that the above example is just a very special case of the following nice fact:
let be a commutative von Neumann regular ring and let there exists a unique such that and
proof. since is a commutative von Neumann regular ring, there exists some such that now let and see that and so we have proved the existence. to prove the uniqueness, suppose that and for some then