TABLE OF CONTENTS This Section is devoted to the notion of the Morita equivalence of rings and Lie groupoids. Throughout this Section, 
is a commutative ring.
Let 
and 
be 
-rings, and let 
and 
denote the categories of left 
- and 
-modules, respectively. These rings are called Morita equivalent if the categories 
and 
are equivalent [52; 78], i.e., there exist quasiinverse functors
| (10.4.1) |  |
Theorems 10.4.1 and 10.4.2 below state that one can construct the functor F (10.4.1) by means of tensor products of 
-bimodules.
Given rings 
, 
and 
, let 
and 
be 
-and 
-bimodules, respectively. Their tensor product
| (10.4.2) |  |
is defined as an additive group generated by elements p ? q, p ? P, q ? Q, modulo the relations
and it is provided with the 
-bimodule structure
Similarly, the tensor product of an 
-bimodule P and a left 
-module Q is a left 
-module, and the tensor product of a right 
-module Q and an 
-bimodule P is a right 
-module.
Any 
)-bimodule 
yields the functor
| (10.4.3) |  |
Furthermore, one can show that, given ( 
)-bimodules P and P', the corresponding functors F P and F P' are equivalent if and only if the bimodules P and P are isomorphic. It follows that, given an 
-bimodule P and a 
-bimodule Q, the functors
are quasi-inverse if and only if there...
