Inabstract algebra,Morita equivalence is a relationship defined betweenrings that preserves manyring-theoretic properties. More precisely, two ringsR,S are Morita equivalent (denoted by${\displaystyle R\approx S}$) if theircategories of modules areadditivelyequivalent (denoted by${\displaystyle {}_{R}M\approx {}_{S}M}$^[a]).^[2] It is named after Japanese mathematicianKiiti Morita who defined equivalence and a similar notion ofduality in 1958.

Rings are commonly studied in terms of theirmodules, as modules can be viewed asrepresentations of rings. Every ringR has a naturalR-module structure on itself where the module action is defined as the multiplication in the ring, so the approach via modules is more general and gives useful information. Because of this, one often studies a ring by studying thecategory of modules over that ring. Morita equivalence takes this viewpoint to a natural conclusion by defining rings to be Morita equivalent if their module categories areequivalent. This notion is of interest only when dealing withnoncommutative rings, since it can be shown that twocommutative rings are Morita equivalent if and only if they areisomorphic.

Two ringsR andS (associative, with 1) are said to be (Morita)equivalent if there is an equivalence of the category of (left) modules overR,R-Mod, and the category of (left) modules overS,S-Mod. It can be shown that the left module categoriesR-Mod andS-Mod are equivalent if and only if the right module categoriesMod-R andMod-S are equivalent. Further it can be shown that anyfunctor fromR-Mod toS-Mod that yields an equivalence is automaticallyadditive.

Any two isomorphic rings are Morita equivalent.

The ring ofn-by-nmatrices with elements inR, denoted M_n R, is Morita-equivalent toR for anyintegern > 0. Notice that this generalizes the classification ofsimpleArtinian rings given byArtin–Wedderburn theory. To see the equivalence, notice that ifX is a leftR-module thenXⁿ is anM_n(R)-module where the module structure is given by matrix multiplication on the left of column vectors fromX. This allows the definition of a functor from the category of leftR-modules to the category of leftM_n(R)-modules. The inverse functor is defined by realizing that for anyM_n(R)-module there is a leftR-moduleX such that theM_n(R)-module is obtained fromX as described above.

Equivalences can be characterized as follows: ifF :R-Mod${\displaystyle \to }$S-Mod andG :S-Mod${\displaystyle \to }$R-Mod are additive (covariant) functors, thenF andG are an equivalence if and only if there is a balanced (S,R)-bimoduleP such that_SP andP_R arefinitely generatedprojectivegenerators and there arenatural isomorphisms of the functors${\displaystyle \mathrm{F}(-)\cong P{\otimes }_R-}$, and of the functors${\displaystyle \mathrm{G}(-)\cong \mathrm{Hom}{(}_{S}P,-).}$ Finitely generated projective generators are also sometimes calledprogenerators for their module category.^[3]

For everyright-exact functorF from the category of leftR-modules to the category of leftS-modules that commutes withdirect sums, a theorem ofhomological algebra shows that there is a (S,R)-bimoduleE such that the functor${\displaystyle \mathrm{F}(-)}$ is naturally isomorphic to the functor${\displaystyle E{\otimes }_R-}$. Since equivalences are by necessity exact and commute with direct sums, this implies thatR andS are Morita equivalent if and only if there are bimodules_RM_S and_SN_R such that${\displaystyle M{\otimes }_{S}N\cong R}$ as (R,R)-bimodules and${\displaystyle N{\otimes }_{R}M\cong S}$ as (S,S)-bimodules. Moreover,N andM are related via an (S,R)-bimodule isomorphism:${\displaystyle N\cong \mathrm{Hom}(M_S,S_S)}$.

More concretely, two ringsR andS are Morita equivalent if and only if${\displaystyle S\cong \mathrm{End}(P_R)}$ for aprogenerator moduleP_R,^[4] which is the case if and only if

${\displaystyle S\cong e{\mathbb{M}}_n(R)e}$

(isomorphism of rings) for some positive integern andfull idempotente in the matrix ring M_n R.

It is known that ifR is Morita equivalent toS, then the ring Z(R) is isomorphic to the ring Z(S), where Z(-) denotes thecenter of the ring, and furthermoreR/J(R) is Morita equivalent toS/J(S), whereJ(-) denotes theJacobson radical.

While isomorphic rings are Morita equivalent, Morita equivalent rings can be nonisomorphic. An easy example is that adivision ringD is Morita equivalent to all of its matrix ringsM_n D, but cannot be isomorphic whenn > 1. In the special case of commutative rings, Morita equivalent rings are actually isomorphic. This follows immediately from the comment above, for ifR is Morita equivalent toS then${\displaystyle R=\mathrm{Z}(R)\cong \mathrm{Z}(S)=S}$.

Many properties are preserved by the equivalence functor for the objects in the module category. Generally speaking, any property of modules defined purely in terms of modules and theirhomomorphisms (and not to their underlying elements or ring) is acategorical property which will be preserved by the equivalence functor. For example, ifF(-) is the equivalence functor fromR-Mod toS-Mod, then theR moduleM has any of the following properties if and only if theS moduleF(M) does:injective,projective,flat,faithful,simple,semisimple,finitely generated,finitely presented,Artinian, andNoetherian. Examples of properties not necessarily preserved include beingfree, and beingcyclic.

Many ring-theoretic properties are stated in terms of their modules, and so these properties are preserved between Morita equivalent rings. Properties shared between equivalent rings are calledMorita invariant properties. For example, a ringR issemisimple if and only if all of its modules are semisimple, and since semisimple modules are preserved under Morita equivalence, an equivalent ringS must also have all of its modules semisimple, and therefore be a semisimple ring itself.

Sometimes it is not immediately obvious why a property should be preserved. For example, using one standard definition ofvon Neumann regular ring (for alla inR, there existsx inR such thata = axa) it is not clear that an equivalent ring should also be von Neumann regular. However another formulation is: a ring is von Neumann regular if and only if all of its modules are flat. Since flatness is preserved across Morita equivalence, it is now clear that von Neumann regularity is Morita invariant.

The following properties are Morita invariant:

• simple,semisimple
• von Neumann regular
• right (or left)Noetherian, right (or left)Artinian
• right (or left)self-injective
• quasi-Frobenius
• prime, right (or left)primitive,semiprime,semiprimitive
• right (or left)(semi-)hereditary
• right (or left)nonsingular
• right (or left)coherent
• semiprimary, right (or left)perfect,semiperfect
• semilocal

Examples of properties which arenot Morita invariant includecommutative,local,reduced,domain, right (or left)Goldie,Frobenius,invariant basis number, andDedekind finite.

There are at least two other tests for determining whether or not a ring property${\displaystyle \mathcal{P}}$ is Morita invariant. An elemente in a ringR is afull idempotent whene² = e andReR = R.

• ${\displaystyle \mathcal{P}}$ is Morita invariant if and only if whenever a ringR satisfies${\displaystyle \mathcal{P}}$, then so doeseRe for every full idempotente and so does every matrix ring M_n R for every positive integern;

or

• ${\displaystyle \mathcal{P}}$ is Morita invariant if and only if: for any ringR and full idempotente inR,R satisfies${\displaystyle \mathcal{P}}$ if and only if the ringeRe satisfies${\displaystyle \mathcal{P}}$.

Dual to the theory of equivalences is the theory ofdualities between the module categories, where the functors used arecontravariant rather than covariant. This theory, though similar in form, has significant differences because there is no duality between the categories of modules for any rings, although dualities may exist forsubcategories. In other words, because infinite-dimensional modules^{[clarification needed]} are not generallyreflexive, the theory of dualities applies more easily to finitely generated algebras over noetherian rings. Perhaps not surprisingly, the criterion above has an analogue for dualities, where the natural isomorphism is given in terms of thehom functor rather than thetensor functor.

Morita equivalence can also be defined in more structured situations, such as for symplecticgroupoids andC*-algebras. In the case of C*-algebras, a stronger type equivalence, calledstrong Morita equivalence, is needed to obtain results useful in applications, because of the additional structure of C*-algebras (coming from theinvolutive *-operation) and also because C*-algebras do not necessarily have an identity element.

If two rings are Morita equivalent, there is an induced equivalence of the respective categories of projective modules since the Morita equivalences will preserveexact sequences (and hence projective modules). Since thealgebraic K-theory of a ring is defined (inQuillen's approach) in terms of thehomotopy groups of (roughly) theclassifying space of thenerve of the (small) category of finitely generated projective modules over the ring, Morita equivalent rings must have isomorphic K-groups.

• Reiner, I. (2003).Maximal Orders. London Mathematical Society Monographs. New Series. Vol. 28.Oxford University Press. pp. 154–169.ISBN 0-19-852673-3.Zbl 1024.16008.

• Module theory
• Ring theory
• Adjoint functors
• Duality theories
• Equivalence (mathematics)

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