Difference between revisions of "Morita equivalence"
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− | An equivalence relation on the class of all [[ring]]s defined as follows: Two rings $R$ and $S$ are called Morita equivalent if the categories of left (right) $R$- and $S$-modules are equivalent. 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. | + | An equivalence relation on the class of all [[ring]]s 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]]. | 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]]. |
Revision as of 20:02, 26 November 2016
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
.
Morita equivalence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Morita_equivalence&oldid=39829