Let be a ring. Write for the category of right modules over . Two rings and are said to be Morita equivalent if and are equivalent as categories (http://planetmath.org/EquivalenceOfCategories). What this means is: we have two functors
such that for any right -module and any right -module , we have
where means that there is an -module isomorphism between and .
Example. Any ring with is Morita equivalent to any matrix ring over it.
Assume . For convenience, we will also say a module to mean a right module.
Next, let be an -module. Write as the matrix whose cell is and everywhere else. For simplicity we write . Note that is an idempotent in : , and commutes with for any : .
Set . For any , define . Since , this multiplication turns into an -module. If is an -module homomorphism, define by . If are -module homomorphisms, then
so that is a covariant functor.
On the other hand, if is any -module, then . Before proving that , let’s do some preliminary work.
Denote by the matrix whose cell is 1 and everywhere else. Then each is idempotent, for , and . From this, we see that , where , and as -modules. Since has an -module structure as we had shown earlier, are all -modules. Let be the projection map, be the embedding of into , and be the isomorphism from to given by . All these are -module homomorphisms since .
Remark. A property in the class of all rings is said to be Morita invariant if, whenever has property and is Morita equivalent to , then has property as well. By the example above, it is clear that commutativity is not a Morita invariant property.
|Date of creation||2013-03-22 16:38:49|
|Last modified on||2013-03-22 16:38:49|
|Last modified by||CWoo (3771)|