Эквивалентность Мориты колец

Equivalence relation on rings
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring theoretic properties. More precisely two rings like R, S are Morita equivalent (denoted by ) if their categories of modules are additively equivalent (denoted by ). It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.

Rings are commonly studied in terms of their modules, as modules can be viewed as representations of rings. Every ring R has a natural R module structure on itself where the module action is defined as the multiplication in the ring, so the approach via modules is more general and gives useful information. Because of this, one often studies a ring by studying the category of modules over that ring. Morita equivalence takes this viewpoint to a natural conclusion by defining rings to be Morita equivalent if their module categories are equivalent. This notion is of interest only when dealing with noncommutative rings, since it can be shown that two commutative rings are Morita equivalent if and only if they are isomorphic.

Two rings R and S (associative, with 1) are said to be (Morita) equivalent if there is an equivalence of the category of (left) modules over R, R Mod, and the category of (left) modules over S, S Mod. It can be shown that the left module categories R Mod and S Mod are equivalent if and only if the right module categories Mod R and Mod S are equivalent. Further it can be shown that any functor from R Mod to S Mod that yields an equivalence is automatically additive.

Any two isomorphic rings are Morita equivalent. The ring of n by n matrices with elements in R, denoted Mn(R), is Morita equivalent to R for any n > 0. Notice that this generalizes the classification of simple artinian rings given by Artin–Wedderburn theory. To see the equivalence, notice that if X is a left R module then Xn is an Mn(R) module where the module structure is given by matrix multiplication on the left of column vectors from X. This allows the definition of a functor from the category of left R modules to the category of left Mn(R) modules. The inverse functor is defined by realizing that for any Mn(R) module there is a left R module X such that the Mn(R) module is obtained from X as described above.

Dual to the theory of equivalences is the theory of dualities between the module categories, where the functors used are contravariant rather than covariant. This theory, though similar in form, has significant differences because there is no duality between the categories of modules for any rings, although dualities may exist for subcategories. In other words, because infinite dimensional modules are not generally reflexive, the theory of dualities applies more easily to finitely generated algebras over noetherian rings. Perhaps not surprisingly, the criterion above has an analogue for dualities, where the natural isomorphism is given in terms of the hom functor rather than the tensor functor. Morita equivalence can also be defined in more structured situations, such as for symplectic groupoids and C* algebras. In the case of C* algebras, a stronger type equivalence, called strong Morita equivalence, is needed to obtain results useful in applications, because of the additional structure of C* algebras (coming from the involutive * operation) and also because C* algebras do not necessarily have an identity element.

If two rings are Morita equivalent, there is an induced equivalence of the respective categories of projective modules since the Morita equivalences will preserve exact sequences (and hence projective modules). Since the algebraic K theory of a ring is defined (in Quillen's approach) in terms of the homotopy groups of (roughly) the classifying space of the nerve of the (small) category of finitely generated projective modules over the ring, Morita equivalent rings must have isomorphic K groups.