Morita equivalence

An equivalence relation on the class of all rings defined as follows: Two rings $R$ and $S$ are called Morita equivalent if the categories of left (right) $R$- and $S$-modules are equivalent (cf. Equivalence of categories). The most important examples of Morita-equivalent rings are: a ring $R$ and the ring of all $n \times n$-matrices over $R$. In order that there is Morita equivalence between two rings $R$ and $S$ it is necessary and sufficient that in the category of left $R$-modules there is a finitely-generated projective generator $U$ such that its ring of endomorphisms is isomorphic to $S$. The left $R$-module $A$ is put in correspondence with the left $S$-module $\mathrm{Hom}_R(U,A)$. Among the properties preserved by transition to a Morita-equivalent ring are the properties of being: Artinian, Noetherian, primary, simple, classically semi-simple, regular, self-injective, hereditary, and primitive.

Alongside with Morita equivalence one considers Morita duality, relating some subcategories of the categories of left $R$-modules and right $S$-modules (mostly the subcategories of finitely-generated modules). However, the very existence of such a duality places definite restrictions on the rings $R$ and $S$. In particular, for $R=S$ this implies that $R$ is a quasi-Frobenius ring.

The general concept of Morita equivalence was developed by K. Morita [1].

References

[1]	K. Morita, Sci. Reports Tokyo Kyoiku Dajkagu A , 6 (1958) pp. 83–142
[2]	H. Bass, "Algebraic $K$-theory" , Benjamin (1968)
[3]	C. Faith, "Algebra: rings, modules and categories" , 1–2 , Springer (1981–1976)
[4]	P. Cohn, "Morita equivalence and duality" , London (1976)

Comments

For generating objects of categories see also Generator of a category.

Let and be categories. A duality is a pair of contravariant functors and such that , , where denotes natural equivalence (functorial isomorphism) and is the identity functor on .

Let and be rings and let and be full subcategories of the categories of right -modules and left -modules , respectively (cf. Module). Let be a bimodule. A duality between and is called a -duality or Morita duality if and are, respectively, naturally equivalent to and . A theorem of Morita says that if and are Abelian full subcategories with and , then any duality between and is a -duality with .