en>Quota: del broken link

2013-11-01T10:34:36Z

del broken link

← Older revision		Revision as of 12:34, 1 November 2013
Line 1:		Line 1:
	~~Alyson Meagher~~ is the ~~title her parents gave her but she doesn~~'~~t like when individuals use her full title~~. ~~Invoicing~~ is ~~my profession~~. The ~~favorite pastime~~ for ~~him~~ and ~~his children~~ is to ~~play lacross~~ and ~~he would~~ by ~~no means give it up~~. ~~My spouse~~ and ~~I live~~ in ~~Mississippi~~ and ~~I love each day residing right here~~.<br><br>~~Also visit my page online psychic reading~~ ([~~http~~://~~ltreme~~.~~com~~/~~index~~.~~php?do~~=/~~profile-127790~~/~~info~~/ ~~ltreme~~.~~com~~])		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.

			== 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]].

			== 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]].

			== 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.

			== 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'' > 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>.

			== 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'' = ''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> = ''e'' and ''ReR'' = ''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.

			== 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 category of projective modules over the ring, Morita equivalent rings must have isomorphic K-groups.

			==References==
			{{reflist}}
			* {{cite journal \| last=Morita \| first=Kiiti \| authorlink=Kiiti Morita \| 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 }}
			* {{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 \| location=[[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 \| 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 }}
			* {{cite book \| last=Lam \| first=T.Y. \| authorlink=T. Y. Lam \| title=A first course in noncommutative rings \| 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 }}
			* {{cite web \| last=Meyer \| first=Ralf \| title=Morita Equivalence In Algebra And Geometry \| url=http://citeseer.ist.psu.edu/meyer97morita.html }}

			[[Category:Module theory\|* Module]]
			[[Category:Ring theory]]
			[[Category:Adjoint functors]]
			[[Category:Duality theories]]

en>Loadmaster: /* Lede */ swap awkward word order

2012-08-20T19:19:37Z

Lede: swap awkward word order

New page

Alyson Meagher is the title her parents gave her but she doesn't like when individuals use her full title. Invoicing is my profession. The favorite pastime for him and his children is to play lacross and he would by no means give it up. My spouse and I live in Mississippi and I love each day residing right here.<br><br>Also visit my page online psychic reading ([http://ltreme.com/index.php?do=/profile-127790/info/ ltreme.com])

← Older revision		Revision as of 12:34, 1 November 2013
Line 1:		Line 1:
	~~Alyson Meagher~~ is the ~~title her parents gave her but she doesn~~'~~t like when individuals use her full title~~. ~~Invoicing~~ is ~~my profession~~. The ~~favorite pastime~~ for ~~him~~ and ~~his children~~ is to ~~play lacross~~ and ~~he would~~ by ~~no means give it up~~. ~~My spouse~~ and ~~I live~~ in ~~Mississippi~~ and ~~I love each day residing right here~~.<br><br>~~Also visit my page online psychic reading~~ ([~~http~~://~~ltreme~~.~~com~~/~~index~~.~~php?do~~=/~~profile-127790~~/~~info~~/ ~~ltreme~~.~~com~~])		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.

			== 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]].

			== 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]].

			== 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.

			== 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'' > 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>.

			== 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'' = ''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> = ''e'' and ''ReR'' = ''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.

			== 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 category of projective modules over the ring, Morita equivalent rings must have isomorphic K-groups.

			==References==
			{{reflist}}
			* {{cite journal \| last=Morita \| first=Kiiti \| authorlink=Kiiti Morita \| 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 }}
			* {{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 \| location=[[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 \| 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 }}
			* {{cite book \| last=Lam \| first=T.Y. \| authorlink=T. Y. Lam \| title=A first course in noncommutative rings \| 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 }}
			* {{cite web \| last=Meyer \| first=Ralf \| title=Morita Equivalence In Algebra And Geometry \| url=http://citeseer.ist.psu.edu/meyer97morita.html }}

			[[Category:Module theory\|* Module]]
			[[Category:Ring theory]]
			[[Category:Adjoint functors]]
			[[Category:Duality theories]]

Decimal representation - Revision history

en>Quota: del broken link

en>Loadmaster: /* Lede */ swap awkward word order