# Morita equivalence

2010 Mathematics Subject Classification: Primary: 16D90 [MSN][ZBL]

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)

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 $U = TA$.