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An equivalence relation on the class of all rings (cf. [[Ring|Ring]]) defined as follows: Two rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m0649301.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m0649302.png" /> are called Morita equivalent if the categories of left (right) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m0649303.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m0649304.png" />-modules are equivalent. The most important examples of Morita-equivalent rings are: a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m0649305.png" /> and the ring of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m0649306.png" />-matrices over it. In order that there is Morita equivalence between two rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m0649307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m0649308.png" /> it is necessary and sufficient that in the [[Category|category]] of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m0649309.png" />-modules there is a finitely-generated projective generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493010.png" /> such that its ring of endomorphisms is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493011.png" />. The left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493012.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493013.png" /> is put in correspondence with the left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493014.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493015.png" />. 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.
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{{TEX|done}}{{MSC|16D90}}
  
Alongside with Morita equivalence one considers Morita duality, relating some subcategories of the categories of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493016.png" />-modules and right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493017.png" />-modules (mostly the subcategories of finitely-generated modules). However, the very existence of such a duality places definite restrictions on the rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493019.png" />. In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493020.png" /> this implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493021.png" /> is a [[Quasi-Frobenius ring|quasi-Frobenius ring]].
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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 (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.
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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]]].
 
The general concept of Morita equivalence was developed by K. Morita [[#References|[1]]].
  
 
====References====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493022.png" />-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>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  K. Morita,  ''Sci. Reports Tokyo Kyoiku Dajkagu A'' , '''6'''  (1958)  pp. 83–142</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  H. Bass,  "Algebraic $K$-theory" , Benjamin  (1968)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules and categories" , '''1–2''' , Springer  (1981–1976)</TD></TR>
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<TR><TD valign="top">[4]</TD> <TD valign="top">  P. Cohn,  "Morita equivalence and duality" , London  (1976)</TD></TR></table>
  
  
  
 
====Comments====
 
====Comments====
For generating objects of categories see also [[Generator of a category|Generator of a category]].
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For generating objects of categories see also [[Generator of a category]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493024.png" /> be categories. A duality is a pair of contravariant functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493029.png" /> denotes natural equivalence (functorial isomorphism) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493030.png" /> is the identity functor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493031.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493033.png" /> be rings and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493035.png" /> be full subcategories of the categories of right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493036.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493037.png" /> and left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493038.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493039.png" />, respectively (cf. [[Module|Module]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493040.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493041.png" /> [[Bimodule|bimodule]]. A duality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493042.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493044.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493046.png" />-duality or Morita duality if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493048.png" /> are, respectively, naturally equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493050.png" />. A theorem of Morita says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493052.png" /> are Abelian full subcategories with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493054.png" />, then any duality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493055.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493057.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493058.png" />-duality with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064930/m06493059.png" />.
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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 $U = TA$.

Latest revision as of 20:17, 26 November 2016

2020 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)


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 $U = TA$.

How to Cite This Entry:
Morita equivalence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Morita_equivalence&oldid=17144
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article