View source for Morita equivalence

{{TEX|done}}{{MSC|16D90}}

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]].

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

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



====Comments====
For generating objects of categories see also [[Generator of a category]].

Let $\mathcal{C}$ and $\mathcal{D}$ be categories. A duality is a pair of contravariant functors $T : \mathcal{D} \rightarrow \mathcal{C}$ and $S : \mathcal{C} \rightarrow \mathcal{D}$ such that $ST \simeq \text{id}_{\mathcal{C}}$, $TS \simeq  \text{id}_{\mathcal{D}}$, where $\simeq$ denotes natural equivalence (functorial isomorphism) and $\text{id}_{\mathcal{X}}$ is the identity functor on $\mathcal{X}$.

Let $A$ and $B$ be rings and let $\mathcal{C}$ and $\mathcal{D}$ be full subcategories of the categories of right $A$-modules $\textbf{Mod}_A$ and left $B$-modules ${}_B\textbf{Mod}$, respectively (cf. [[Module]]). Let $U$ be a $(B,A)$ [[bimodule]]. A duality $(T,S)$ between $\mathcal{C}$ and $\mathcal{D}$ is called a $U$-duality or Morita duality if $T$ and $S$ are, respectively, naturally equivalent to $\text{Hom}_A({\cdot},U)$ and $\text{Hom}_B(U,{\cdot})$. A theorem of Morita says that if $\mathcal{C}$ and $\mathcal{D}$ are Abelian full subcategories with $A \in \mathcal{C}$ and $B \in \mathcal{D}$, then any duality $(T,S)$ between $\mathcal{C}$ and $\mathcal{D}$ is a $U$-duality with $U = TA$.