森田同値のソースを表示

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In [[abstract algebra]], '''Morita equivalence''' is a relationship defined between [[ring (mathematics)|rings]] that preserves many ring-theoretic properties. It is named after Japanese mathematician [[Kiiti Morita]] who defined equivalence and a similar notion of duality in 1958.
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[[代数学]]において、'''森田同値'''（もりたどうち、{{lang-en-short|Morita equivalence}}）とは、[[環論]]的な多くの性質を保つ[[環 (数学)|環]]の間の関係のことを言う。これは{{harvtxt|Morita|1958}}において同値関係と双対性に関する記号を定義した[[森田紀一]]にちなんで名付けられた。

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== Motivation ==
[[ring (mathematics)|Rings]] are commonly studied in terms of their [[module (mathematics)|modules]], as modules can be viewed as [[representation theory|representations]] of rings. Every ring ''R'' has a natural ''R''-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 the [[category (mathematics)|category]] 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 are [[Equivalence of categories|equivalent in the sense of category theory]].

This notion is of interest only when dealing with [[noncommutative ring]]s, since it can be shown that two [[commutative ring]]s are Morita equivalent if and only if they are [[Glossary_of_ring_theory#Homomorphisms_and_ideals|isomorphic]].
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== 動機 ==
[[環 (数学)|環]]はその[[環上の加群]]を通じて研究されることが一般的である。これは[[加群]]が環の[[表現論|表現]]と見做せるからである。すべての環 {{mvar|R}} は環の積による作用によって自然に {{mvar|R}} 加群の構造を持つので、加群論的な研究方法はより一般的で有益な情報をもたらす。このような訳で、環についての研究はその環上の加群の成す[[圏 (数学)|圏]]を研究することによってしばしば為される。

この視点からの自然な帰結として、環が森田同値であるとはその環上の加群の成す圏が[[圏同値]]であることと定めた。

この表記方法は[[非可換環]]を扱っている場合にのみ興味の対象となる。なぜなら[[可換環]]が森田同値である必要十分条件は環同型であるからである。これは一般に森田同値な[[環の中心]]が環同型なことから従う。

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== Definition ==
Two rings ''R'' and ''S'' (associative, with 1) are said to be ('''Morita''') '''equivalent''' if there is an equivalence of the category of (left) modules over ''R'', ''R-Mod'', and the category of (left) modules over ''S'', ''S-Mod''. It can be shown that the left module categories ''R-Mod'' and ''S-Mod'' are equivalent if and only if the right module categories ''Mod-R'' and ''Mod-S'' are equivalent. Further it can be shown that any functor from ''R-Mod'' to ''S-Mod'' that yields an equivalence is automatically [[Additive functor|additive]].
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== 定義 ==
（結合的で単位元を持つ）環 {{math|''R'', ''S''}} が（'''森田'''）'''同値'''であるとは、（左）{{mvar|R}} 加群の成す圏 {{math|''R''-Mod}} と（左）{{mvar|S}} 加群の成す圏 {{math|''S''-Mod}} との間に[[圏同値]]があることを言う。左加群の成す圏 {{math|''R''-Mod}} と {{math|''S''-Mod}} とが圏同値である必要十分条件は、右加群の成す圏 {{math|Mod-''R''}} と {{math|Mod-''S''}} とが圏同値であることを示すことができる{{sfn|Anderson|Fuller|1992|loc=Corollary 22.3}}。さらに圏同値を与えるどんな {{math|''R''-Mod}} から {{math|''S''-Mod}} への[[関手]]も自動的に[[前加法圏#加法的関手|加法的]]であることを示すことができる。

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== Examples ==
Any two isomorphic rings are Morita equivalent.

The ring of ''n''-by-''n'' [[matrix (mathematics)|matrices]] with elements in ''R'', denoted ''M''<sub>''n''</sub>(''R''), is Morita-equivalent to ''R'' for any ''n > 0''. Notice that this generalizes the classification of simple artinian rings given by [[Artin–Wedderburn theorem|Artin–Wedderburn theory]]. To see the equivalence, notice that if  ''M'' is a left ''R''-module then ''M<sup>n</sup>'' is an M<sub>n</sub>(''R'')-module where the module structure is given by matrix multiplication on the left of column vectors from ''M''. This allows the definition of a functor from the category of left ''R''-modules to the category of left M<sub>n</sub>(''R'')-modules. The inverse functor is defined by realizing that for any M<sub>n</sub>(''R'')-module there is a left ''R''-module ''V'' and a positive integer ''n'' such that the M<sub>n</sub>(''R'')-module is obtained from ''V'' as described above.
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== 例 ==
同型な環は森田同値である。

任意の環 {{mvar|R}} と非負整数 {{mvar|n}} について {{mvar|R}} 成分の {{mvar|n}} 次[[正方行列]]から成る[[全行列環]] {{math|M<sub>''n''</sub>(''R'')}} は環 {{mvar|R}} と森田同値である{{sfn|Anderson|Fuller|1992|loc=Colorally 22.6}}。これは[[アルティン・ウェダーバーンの定理|アルティン‐ウェダーバーン理論]]によって与えられる[[単純環|単純]][[アルティン環]]の分類の一般化になっていることに注意する。森田同値を確かめるには、もし {{mvar|M}} が左 {{mvar|R}} 加群ならば {{math|''M<sup>n</sup>''}} は行ベクトルに対する左から行列の掛け算によって {{math|M<sub>''n''</sub>(''R'')}} 加群の構造が与えられることに注意すればよい。これは左 {{mvar|R}} 加群の圏 {{math|''R''-Mod}} から左 {{math|M<sub>''n''</sub>(''R'')}} 加群の圏 {{math|M<sub>''n''</sub>(''R'')-Mod}} への関手を定める。

== 同値の判定法 ==
森田同値は次のように特徴付けられる{{sfn|Anderson|Fuller|1992|loc=Theorem 22.2}}。もし {{math|''F'' : ''R''-Mod &rarr; ''S''-Mod}} と {{math|''G'' : ''S''-Mod &rarr; ''R''-Mod}} が加法的（共変）[[関手]]ならば、{{math|''F'', ''G''}} が森田同値を定める必要十分条件は、ある平衡 {{math|(''S'', ''R'')}} 両側加群 {{mvar|P}} が存在して {{math|''<sub>S</sub>P''}} と {{math|''P<sub>R</sub>''}} が[[有限生成加群|有限生成]][[射影加群|射影的]][[生成素]]で、
さらに関手の[[自然変換#定義|自然同型]] {{math|''F''(&ndash;) &cong; ''P'' &otimes;<sub>''R''</sub> &ndash;}} と {{math|''G''(&ndash;) &cong; Hom(''<sub>S</sub>P'', &ndash;)}} が存在することである。有限生成射影的生成素はその加群の圏の'''射影生成素'''（{{lang-en-short|progenerators}}）と呼ばれることもある<ref name=Sep6>DeMeyer & Ingraham (1971) p.6</ref>。

左 {{mvar|R}} 加群の圏から左 {{mvar|S}} 加群の圏への[[加群の直和|直和]]と可換なすべての[[完全関手|右完全関手]] {{mvar|F}} に対して、Eilenberg-Wattsの定理よりある {{math|(''S'', ''R'')}} 両側加群 {{mvar|E}} が存在して、関手 {{math|''F''(&ndash;)}} は関手 {{math|''E'' &otimes;<sub>''R''</sub> &ndash;}} と自然同型である<ref>{{cite web
|url=https://ncatlab.org/nlab/show/Eilenberg-Watts+theorem
|title=Eilenberg-Watts theorem
|work=[[nLab]]
|accessdate=2019-04-20
}}</ref>。同値は完全で直和と可換なことが必要なので、このことは {{mvar|R}} と {{mvar|S}} が森田同値である必要十分条件はある両側加群 {{math|''<sub>R</sub>M<sub>S</sub>''}} と {{math|''<sub>S</sub>N<sub>R</sub>''}} が存在して、
{{math|(''R'', ''R'')}} 両側加群としての同型 {{math|''M'' &otimes;<sub>''S''</sub> ''N'' &cong; ''R''}} と {{math|(''S'', ''S'')}} 両側加群としての同型 {{math|''N'' &otimes;<sub>''R''</sub> ''M'' &cong; ''S''}} が成り立つことを示している。さらに {{mvar|N}} と {{mvar|M}} は {{math|(''S'', ''R'')}} 両側加群としての同型 {{math|''N'' &cong; Hom(''M<sub>S</sub>'', ''S<sub>S</sub>'')}} によって関連づけられる。
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== Criteria for equivalence ==
Equivalences can be characterized as follows: if ''F'':''R-Mod'' <math>\to</math> ''S-Mod'' and ''G'':''S-Mod''<math>\to</math> ''R-Mod'' are additive (covariant) [[functors]], then ''F'' and ''G'' are an equivalence if and only if there is a balanced (''S'',''R'')-[[bimodule]] ''P'' such that  <sub>S</sub>''P''  and  ''P''<sub>R</sub>  are [[finitely generated module|finitely generated]] [[projective module|projective]] [[generator (category theory)|generators]] and there are [[natural transformation|natural isomorphisms]] of the functors <math> \operatorname{F}(-) \cong P \otimes_R - </math>, and of the functors <math>\operatorname{G}(-) \cong \operatorname{Hom}(_{S}P,-).</math> Finitely generated projective generators are also sometimes called '''progenerators''' for their module category.<ref name=Sep6>DeMeyer & Ingraham (1971) p.6</ref>

For every [[exact functor|right-exact]] functor ''F'' from the category of left-''R'' modules to the category of left-''S'' modules that commutes with [[direct sum]]s, a theorem of [[homological algebra]] shows that there is a ''(S,R)''-bimodule ''E'' such that the functor <math>\operatorname{F}(-)</math> is naturally isomorphic to the functor <math>E \otimes_R -</math>. Since equivalences are by necessity exact and commute with direct sums, this implies that ''R'' and ''S'' are Morita equivalent if and only if there are bimodules ''<sub>R</sub>M<sub>S</sub>'' and ''<sub>S</sub>N<sub>R</sub>'' such that <math>M \otimes_{S} N \cong R</math> as ''(R,R)'' bimodules and <math>N \otimes_{R} M \cong S</math> as ''(S,S)'' bimodules.  Moreover, ''N'' and ''M'' are related via an ''(S,R)'' bimodule isomorphism: <math>N \cong \operatorname{Hom}(M_S,S_S)</math>.

More concretely, two rings ''R'' and ''S'' are Morita equivalent if and only if <math>S\cong \operatorname{End}(P_R)</math> for a [[progenerator]] module ''P<sub>R</sub>'',<ref name=Sep16>DeMeyer & Ingraham (1971) p.16</ref> which is the case if and only if 
:<math>S\cong e\mathbb{M}_{n}(R)e</math> 
(isomorphism of rings) for some positive integer ''n'' and [[Idempotent_element#Types_of_ring_idempotents|full idempotent]] ''e'' in the matrix ring M<sub>n</sub>(''R'').

It is known that if ''R'' is Morita equivalent to ''S'', then the ring C(''R'') is isomorphic to the ring C(''S''), where C(-) denotes the [[center of a ring|center of the ring]], and furthermore ''R''/''J''(''R'') is Morita equivalent to ''S''/''J''(''S''), where ''J''(-) denotes the [[Jacobson radical]].

While isomorphic rings are Morita equivalent, Morita equivalent rings can be nonisomorphic.  An easy example is that a [[division ring]] ''D'' is Morita equivalent to all of its matrix rings ''M''<sub>''n''</sub>(''D''), but cannot be isomorphic when ''n''&nbsp;>&nbsp;1.  In the special case of commutative rings, Morita equivalent rings are actually isomorphic.  This follows immediately from the comment above, for if ''R'' is Morita equivalent to ''S'', <math>R=\operatorname{C}(R)\cong \operatorname{C}(S)=S</math>.
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== 同値不変な性質 ==
多くの性質が加群の圏の対象による森田同値を与える関手によって保たれる。一般的に、（台集合の元や環に依らずに）加群とその準同型のみで定義される加群の性質は、森田同値を与える関手によって保たれる'''圏論的性質'''である。たとえば {{math|''F''(&ndash;)}} が {{math|''R''-Mod}} から {{math|''S''-Mod}} への森田同値を与える関手ならば、{{mvar|R}} 加群 {{mvar|M}} が次の性質をもつ必要十分条件は {{mvar|S}} 加群 {{math|''F''(''M'')}} がその性質を持つことである：[[入射加群|入射的]]・[[射影加群|射影的]]・[[平坦加群|平坦]]・[[有限生成加群|有限生成]]・[[有限生成加群#有限表示、有限関係、連接加群|有限表示的]]・[[アルティン加群|アルティン的]]・[[ネーター加群|ネーター的]]。森田同値不変とは限らない性質には[[自由加群|自由]]であることや[[巡回加群|巡回的]]であることがある。

多くの環論的性質はその環上の加群のことばで述べられるので、これらの性質は森田同値な環の間で保たれる。森田同値な環で共有される性質は'''森田不変量'''と呼ばれる。たとえば環 {{mvar|R}} が[[半単純環]]である必要十分条件はその環上のすべての加群が[[半単純加群]]であることで、加群の半単純性は森田同値で保たれるので、森田同値な環 {{mvar|S}} 上の加群もすべて半単純であり、したがって {{mvar|S}} も半単純環である。ある性質がなぜ保たれなければならないのかが明らかではないこともある。たとえば標準的な[[フォン・ノイマン正則環]]の定義（すべての {{mvar|R}} の元 {{mvar|a}} に対して {{mvar|R}} の元 {{mvar|x}} が存在して、{{math|''a'' {{=}} ''axa''}} を満たす）の下で森田同値な環もフォン・ノイマン正則環でなければならないことは明らかではない。しかし他の定式化がある：環がフォン・ノイマン正則環である必要十分条件は、その環上の加群がすべて[[平坦加群|平坦]]であることである。平坦性は森田同値で保たれるので、フォン・ノイマン正則性が森田不変量であることがわかった。

以下の性質は森田不変量である。
{{Div col|cols=2}}
* [[単純環|単純]]、[[半単純環|半単純]]<ref name="Cor 21.9">{{harvnb|Anderson|Fuller|1992|loc=Corollary 21.9.}}</ref>
* [[フォン・ノイマン正則環|フォン・ノイマン正則性]]<ref name="Ex 21.12">{{harvnb|Anderson|Fuller|1992|loc=Exercise 21.12.}}</ref>
* 左（あるいは右）[[ネーター環|ネーター性]]、左（あるいは右）[[アルティン環|アルティン性]]<ref name="Cor 21.9" />
* 左（あるいは右）[[入射加群#自己移入環|自己入射的]]<ref name="Ex 21.12" />
* [[準フロベニウス環|準フロベニウス的]]
* [[素環|素]]、左（あるいは右）[[原始環|原始的]]<ref name="Cor 21.9" />、[[半素環|半素]]、[[半原始環|半原始的]]
* 左（あるいは右）[[遺伝環|遺伝的]]<ref name="Ex 21.12" />
* 左（あるいは右）[[非特異環|非特異]]
* 左（あるいは右）[[連接環|連接]]
* [[半準素環|半準素]]、左（あるいは右）[[完全環|完全]]、[[半完全環|半完全]]
* [[半局所環|半局所]]
{{Div col end}}
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== Properties preserved by equivalence ==
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 their homomorphisms (and not to their underlying elements or ring) is a '''categorical property''' which will be preserved by the equivalence functor.  For example, if ''F''(-) is the equivalence functor from ''R-Mod'' to ''S-Mod'', then the ''R'' module ''M'' has any of the following properties if and only if the ''S'' module ''F''(''M'') does: [[injective module|injective]], [[projective module|projective]], [[flat module|flat]], [[faithful module|faithful]], [[simple module|simple]], [[semisimple module|semisimple]], [[finitely generated module|finitely generated]], [[Finitely-generated_module#Finitely_presented.2C_finitely_related.2C_and_coherent_modules|finitely presented]], [[artinian module|Artinian]], and [[noetherian module|Noetherian]].  Examples of properties not necessarily preserved include being [[free module|free]], and being [[cyclic module|cyclic]].

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 called '''Morita invariant''' properties.  For example, a ring ''R'' is [[Semisimple_module#Semisimple_rings|semisimple]] if and only if all of its modules are semisimple, and since semisimple modules are preserved under Morita equivalence, an equivalent ring ''S'' 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 of [[von Neumann regular ring]] (for all ''a'' in ''R'', there exists ''x'' in ''R'' such that ''a''&nbsp;=&nbsp;''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 ring|simple]], [[Semisimple ring|semisimple]]
*[[von Neumann regular ring|von Neumann regular]]
*right (or left) [[Noetherian ring|Noetherian]], right (or left) [[Artinian ring|Artinian]]
*right (or left) [[Injective_module#Self-injective_rings|self-injective]]
*[[quasi-Frobenius ring|quasi-Frobenius]]
*[[prime ring|prime]], right (or left) [[Primitive ring|primitive]], [[semiprime ring|semiprime]], [[semiprimitive ring|semiprimitive]]
*right (or left) [[hereditary ring|(semi-)hereditary]]
*right (or left) [[nonsingular ring|nonsingular]]
*right (or left) [[coherent ring|coherent]]
*[[semiprimary ring|semiprimary]], right (or left) [[perfect ring|perfect]], [[semiperfect ring|semiperfect]] 
*[[semilocal ring|semilocal]]

Examples of properties which are ''not'' Morita invariant include [[commutative ring|commutative]], [[local ring|local]], [[reduced ring|reduced]], [[domain (ring theory)|domain]], right (or left) [[Goldie ring|Goldie]], [[Frobenius ring|Frobenius]], [[invariant basis number]], and [[Dedekind finite ring|Dedekind finite]].

There are at least two other tests for determining whether or not a ring property <math>\mathcal{P}</math> is Morita invariant.  An element ''e'' in a ring ''R'' is a '''full idempotent''' when ''e''<sup>2</sup>&nbsp;=&nbsp;''e'' and ''ReR''&nbsp;=&nbsp;''R''.

*<math>\mathcal{P}</math> is Morita invariant if and only if whenever a ring ''R'' satisfies <math>\mathcal{P}</math>, then so does ''eRe'' for every full idempotent ''e'' and so does every matrix ring M<sub>n</sub>(''R'') for every positive integer ''n'';
or
*<math>\mathcal{P}</math> is Morita invariant if and only if: for any ring ''R'' and full idempotent ''e'' in ''R'', ''R'' satisfies <math>\mathcal{P}</math> if and only if the ring ''eRe'' satisfies <math>\mathcal{P}</math>.

== Further directions ==
Dual to the theory of equivalences is the theory of [[Opposite category|dualities]] between the module categories, where the functors used are [[Functor#Covariance_and_contravariance|contravariant]] 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 for subcategories. In other words, because infinite dimensional modules are not generally [[reflexive space|reflexive]], 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 the hom functor rather than the tensor functor.

Morita Equivalence can also be defined in more structured situations, such as for symplectic groupoids and [[C*-algebra]]s. In the case of C*-algebras, a stronger type equivalence, called '''strong Morita equivalence''', is needed to obtain results useful in applications, because of the additional structure of C*-algebras (coming from the involutive *-operation) and also because C*-algebras do not necessarily have an identity element.

== Significance in K-theory ==
If two rings are Morita equivalent, there is an induced equivalence of the respective categories of projective modules since the Morita equivalences will preserve exact sequences (and hence projective modules). Since the [[algebraic K-theory]] of a ring is defined (in Quillen's approach) in terms of the [[homotopy group]]s of the [[classifying space]] of the [[nerve (category theory)|nerve]] of the (small) category of finitely generated projective modules over the ring, Morita equivalent rings must have isomorphic K-groups.

==References==
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== 脚注 ==
{{reflist|2}}

== 参考文献 ==
* {{cite journal | last=Morita | first=Kiiti | authorlink=森田紀一 | title=Duality for modules and its applications to the theory of rings with minimum condition | journal=Science reports of the Tokyo Kyoiku Daigaku. Section A | year=1958 | volume=6 | issue=150 | pages=83–142 | zbl=0080.25702 | issn=0371-3539 | ref=harv }}
* {{cite book | last1=DeMeyer | first1=F. | last2=Ingraham | first2=E. | title=Separable algebras over commutative rings | series=Lecture Notes in Mathematics | volume=181 | location=Berlin-Heidelberg-New York | publisher=[[Springer-Verlag]] | year=1971 | isbn=978-3-540-05371-2 | zbl=0215.36602 }}
* {{cite book | first1=F.W. | last1=Anderson | first2=K.R. | last2=Fuller | title=Rings and Categories of Modules | url={{google books|MALaBwAAQBAJ|Rings and Categories of Modules|plainurl=yes}} | series=[[Graduate Texts in Mathematics]] | volume=13 | edition=2nd | publisher=[[Springer-Verlag]] | location=New York | year=1992 | isbn=0-387-97845-3 | zbl=0765.16001 | ref=harv }}
* {{cite book | last=Lam | first=T.Y. | title=A first course in noncommutative rings | url={{google books|2T5DAAAAQBAJ|A first course in noncommutative rings|plainurl=yes}} | edition=2nd | series=[[Graduate Texts in Mathematics]] | volume=131 | location=New York, NY | publisher=[[Springer-Verlag]] | year=2001 | isbn=0-387-95183-0 | at=Chapters 17-18-19 | zbl=0980.16001 }}
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* {{cite web | last=Meyer | first=Ralf | title=Morita Equivalence In Algebra And Geometry | url=http://citeseer.ist.psu.edu/meyer97morita.html }}
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==Further reading==
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* {{cite book | last=Reiner | first=I. | title=Maximal Orders | series=London Mathematical Society Monographs.  New Series | volume=28 | publisher=[[Oxford University Press]] | year=2003 | isbn=0-19-852673-3 | zbl=1024.16008 | pages=154-169 }}

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