In mathematics, especially in the area of abstract algebra, every module has an associated character module. Using the associated character module it is possible to investigate the properties of the original module. One of the main results discovered by Joachim Lambek shows that a module is flat if and only if the associated character module is injective.[1]
The group, the group of rational numbers modulo , can be considered as a -module in the natural way. Let be an additive group which is also considered as a -module. Then the group
of -homomorphisms from to is called the character group associated to . The elements in this group are called characters. If is a left -module over a ring , then the character group is a right -module and called the character module associated to. The module action in the character module for and is defined by for all .[2] The character module can also be defined in the same way for right -modules. In the literature also the notations and are used for character modules.[3][4]
Let be left -modules and an -homomorphismus. Then the mapping defined by for all is a right -homomorphism. Character module formation is a contravariant functor from the category of left -modules to the category of right -modules.[3]
Motivationedit
The abelian group is divisible and therefore an injective-module. Furthermore it has the following important property: Let be an abelian group and nonzero. Then there exists a group homomorphism with . This says that is a cogenerator. With these properties one can show the main theorem of the theory of character modules:[3]
Theorem (Lambek)[1]: A left module over a ring is flat if and only if the character module is an injective right -module.
Propertiesedit
Let be a left module over a ring and the associated character module.
The module is flat if and only if is injective (Lambek's Theorem[4]).[1]
If is free, then is an injective right -module and is a direct product of copies of the right -modules .[2]
For every right -module there is a free module such that is isomorphic to a submodule of . With the previous property this module is injective, hence every right -module is isomorphic to a submodule of an injective module. (Baer's Theorem)[5]
A left -module is injective if and only if there exists a free such that is isomorphic to a direct summand of .[5]
The module is injective if and only if it is a direct summand of a character module of a free module.[2]
If is a submodule of , then is isomorphic to the submodule of which consists of all elements which annihilate .[2]
Character module formation is a contravariant exact functor, i.e. it preserves exact sequences.[3]
Let be a right -module. Then the modules and are isomorphic as -modules.[4]
Referencesedit
^ abcLambek, Joachim (1964). "A Module is Flat if and Only if its Character Module is Injective". Canadian Mathematical Bulletin. 7 (2): 237–243. doi:10.4153/CMB-1964-021-9. ISSN 0008-4395.
^ abcdLambek, Joachim. (2009). Lectures on rings and modules. American Mathematical Society. Providence, RI: AMS Chelsea Pub. ISBN9780821849002. OCLC 838801039.
^ abcdLam, Tsit-Yuen (1999). Lectures on Modules and Rings. Graduate Texts in Mathematics. Vol. 189. New York, NY: Springer New York.
^ abcTercan, Adnan; Yücel, Canan C. (2016). Module theory, extending modules and generalizations. Frontiers in Mathematics. Switzerland: Birkhäuser. ISBN9783034809528.
^ abBehrens, Ernst-August. (1972). Ring theory. New York: Academic Press. ISBN9780080873572. OCLC 316568566.
April 14, 2024
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In mathematics especially in the area of abstract algebra every module has an associated character module Using the associated character module it is possible to investigate the properties of the original module One of the main results discovered by Joachim Lambek shows that a module is flat if and only if the associated character module is injective 1 Contents 1 Definition 2 Motivation 3 Properties 4 ReferencesDefinition editThe group Q Z displaystyle mathbb Q mathbb Z nbsp the group of rational numbers modulo 1 textstyle 1 nbsp can be considered as a Z displaystyle mathbb Z nbsp module in the natural way Let M displaystyle M nbsp be an additive group which is also considered as a Z displaystyle mathbb Z nbsp module Then the groupM HomZ M Q Z displaystyle M operatorname Hom mathbb Z M mathbb Q mathbb Z nbsp of Z displaystyle mathbb Z nbsp homomorphisms from M displaystyle M nbsp to Q Z displaystyle mathbb Q mathbb Z nbsp is called the character group associated to M displaystyle M nbsp The elements in this group are called characters If M displaystyle M nbsp is a left R displaystyle R nbsp module over a ring R displaystyle R nbsp then the character group M displaystyle M nbsp is a right R displaystyle R nbsp module and called the character module associated to M displaystyle M nbsp The module action in the character module for f HomZ M Q Z displaystyle f in operatorname Hom mathbb Z M mathbb Q mathbb Z nbsp and r R displaystyle r in R nbsp is defined by fr m f rm displaystyle fr m f rm nbsp for all m M displaystyle m in M nbsp 2 The character module can also be defined in the same way for right R displaystyle R nbsp modules In the literature also the notations M M0 displaystyle M M 0 nbsp and M displaystyle M nbsp are used for character modules 3 4 Let M N displaystyle M N nbsp be left R displaystyle R nbsp modules and f M N displaystyle f colon M to N nbsp an R displaystyle R nbsp homomorphismus Then the mapping f N M displaystyle f colon N to M nbsp defined by f h h f displaystyle f h h circ f nbsp for all h N displaystyle h in N nbsp is a right R displaystyle R nbsp homomorphism Character module formation is a contravariant functor from the category of left R displaystyle R nbsp modules to the category of right R displaystyle R nbsp modules 3 Motivation editThe abelian group Q Z displaystyle mathbb Q mathbb Z nbsp is divisible and therefore an injective Z displaystyle mathbb Z nbsp module Furthermore it has the following important property Let G displaystyle G nbsp be an abelian group and g G displaystyle g in G nbsp nonzero Then there exists a group homomorphism f G Q Z displaystyle f colon G to mathbb Q mathbb Z nbsp with f g 0 displaystyle f g neq 0 nbsp This says that Q Z displaystyle mathbb Q mathbb Z nbsp is a cogenerator With these properties one can show the main theorem of the theory of character modules 3 Theorem Lambek 1 A left module M displaystyle M nbsp over a ring R displaystyle R nbsp is flat if and only if the character module M displaystyle M nbsp is an injective right R displaystyle R nbsp module Properties editLet M displaystyle M nbsp be a left module over a ring R displaystyle R nbsp and M displaystyle M nbsp the associated character module The module M displaystyle M nbsp is flat if and only if M displaystyle M nbsp is injective Lambek s Theorem 4 1 If M displaystyle M nbsp is free then M displaystyle M nbsp is an injective right R displaystyle R nbsp module and M displaystyle M nbsp is a direct product of copies of the right R displaystyle R nbsp modules R displaystyle R nbsp 2 For every right R displaystyle R nbsp module N displaystyle N nbsp there is a free module M displaystyle M nbsp such that N displaystyle N nbsp is isomorphic to a submodule of M displaystyle M nbsp With the previous property this module M displaystyle M nbsp is injective hence every right R displaystyle R nbsp module is isomorphic to a submodule of an injective module Baer s Theorem 5 A left R displaystyle R nbsp module N displaystyle N nbsp is injective if and only if there exists a free M displaystyle M nbsp such that N displaystyle N nbsp is isomorphic to a direct summand of M displaystyle M nbsp 5 The module M displaystyle M nbsp is injective if and only if it is a direct summand of a character module of a free module 2 If N displaystyle N nbsp is a submodule of M displaystyle M nbsp then M N displaystyle M N nbsp is isomorphic to the submodule of M displaystyle M nbsp which consists of all elements which annihilate N displaystyle N nbsp 2 Character module formation is a contravariant exact functor i e it preserves exact sequences 3 Let N displaystyle N nbsp be a right R displaystyle R nbsp module Then the modules HomR N M displaystyle operatorname Hom R N M nbsp and N RM displaystyle N otimes R M nbsp are isomorphic as Z displaystyle mathbb Z nbsp modules 4 References edit a b c Lambek Joachim 1964 A Module is Flat if and Only if its Character Module is Injective Canadian Mathematical Bulletin 7 2 237 243 doi 10 4153 CMB 1964 021 9 ISSN 0008 4395 a b c d Lambek Joachim 2009 Lectures on rings and modules American Mathematical Society Providence RI AMS Chelsea Pub ISBN 9780821849002 OCLC 838801039 a b c d Lam Tsit Yuen 1999 Lectures on Modules and Rings Graduate Texts in Mathematics Vol 189 New York NY Springer New York a b c Tercan Adnan Yucel Canan C 2016 Module theory extending modules and generalizations Frontiers in Mathematics Switzerland Birkhauser ISBN 9783034809528 a b Behrens Ernst August 1972 Ring theory New York Academic Press ISBN 9780080873572 OCLC 316568566 Retrieved from https en wikipedia org w index php title Character module amp oldid 1138657262, wikipedia, wiki, book, books, library,