I have never seen a name for this. I think, if it hasn't been defined before, I would be inclined to call it the Morita group. There is a large groupoid, let me call it the Morita groupoid, whose objects are rings and for which a morphism R --> S is a left S, right R bimodule M such that tensoring with M gives an equivalence between the category of left R modules and left S modules. This is locally small since M must be a finitely generated projective left S module and the group you are dealing with is simply the group of endomorphisms of R in that groupoid. the whole theory is due to Morita (and the main theorem, the Morita theorem). This is for rings, of course. I assume that a ringoid is a small preadditive category. A preadditive category with finitely many objects is Morita equivalent to a ring so it will be true for them. Beyond that, it would have to be examined because I am not sure what corresponds to finitely generated. On Sat, 16 Dec 2000, Bill Rowan wrote:

Tensor product gives a monoid structure on the class of isomorphism types of R,R-bimodules, for a ring or ringoid R. Restricting to those elements for which there is a two-sided inverse yields a group. I am inclined to call this the nonabelian Picard group and denote it by NPic(R). If we start with a commutative ring R, then the usual Picard group of R, Pic(R), can be viewed as an abelian subgroup of NPic(R).

Has anyone seen this before? Does anyone have some other idea about what this should be called?

Bill Rowan