In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties.
[2] It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.
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.
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.
Further it can be shown that any functor from R-Mod to S-Mod that yields an equivalence is automatically additive.
The ring of n-by-n matrices with elements in R, denoted Mn R, is Morita-equivalent to R for any integer n > 0.
Notice that this generalizes the classification of simple artinian rings given by Artin–Wedderburn theory.
R-Mod are additive (covariant) functors, then F and G are an equivalence if and only if there is a balanced (S,R)-bimodule P such that SP and PR are finitely generated projective generators and there are natural isomorphisms of the functors
[3] For every right-exact functor F from the category of left R-modules to the category of left S-modules that commutes with direct sums, a theorem of homological algebra shows that there is a (S,R)-bimodule E such that the functor
Many properties are preserved by the equivalence functor for the objects in the module category.
Generally speaking, any property of modules defined purely in terms of modules and their homomorphisms (and not to their underlying elements or ring) is a categorical property which will be preserved by the equivalence functor.
For example, if F(-) is the equivalence functor from R-Mod to S-Mod, then the R module M has any of the following properties if and only if the S module F(M) does: injective, projective, flat, faithful, simple, semisimple, finitely generated, finitely presented, Artinian, and Noetherian.
Examples of properties not necessarily preserved include being free, and being cyclic.
For example, using one standard definition of von Neumann regular ring (for all a in R, there exists x in R such that a = axa) it is not clear that an equivalent ring should also be von Neumann regular.
However another formulation is: a ring is von Neumann regular if and only if all of its modules are flat.
An element e in a ring R is a full idempotent when e2 = e and ReR = R. or Dual to the theory of equivalences is the theory of dualities between the module categories, where the functors used are contravariant rather than covariant.
In other words, because infinite-dimensional modules[clarification needed] are not generally reflexive, the theory of dualities applies more easily to finitely generated algebras over noetherian rings.
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.
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.