Picard group of a ringoid

Picard group of a ringoid

[Bill Rowan]

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

Re: Picard group of a ringoid

[Michael Barr]

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
>