V-ring (ring theory): Difference between revisions
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In [[mathematics]], a '''V-ring''' is a [[ring (mathematics)|ring]] ''R'' such that every [[simple module|simple]] ''R''-[[module (mathematics)|module]] is [[injective module|injective]]. The following three conditions are equivalent:<ref>{{cite book|last=Faith|first=Carl|year=1973|title=Algebra: Rings, modules, and categories|publisher=Springer-Verlag|url=https://books.google.com/books?id=SADvAAAAMAAJ|isbn=978-0387055510|access-date=24 October 2015}}</ref> |
In [[mathematics]], a '''V-ring''' is a [[ring (mathematics)|ring]] ''R'' such that every [[simple module|simple]] ''R''-[[module (mathematics)|module]] is [[injective module|injective]]. The following three conditions are equivalent:<ref>{{cite book|last=Faith|first=Carl|year=1973|title=Algebra: Rings, modules, and categories|publisher=Springer-Verlag|url=https://books.google.com/books?id=SADvAAAAMAAJ|isbn=978-0387055510|access-date=24 October 2015}}</ref> |
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#Every simple left ( |
#Every simple left (respectively right) ''R''-module is injective. |
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#The [[radical of a module|radical]] of every left ( |
#The [[radical of a module|radical]] of every left (respectively right) ''R''-module is zero. |
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#Every left ( |
#Every left (respectively right) [[ideal (ring theory)|ideal]] of ''R'' is an intersection of [[maximal ideal|maximal]] left (respectively right) ideals of ''R''. |
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A [[commutative ring]] is a V-ring [[if and only if]] it is [[Von Neumann regular ring|Von Neumann regular]].<ref>{{cite journal|title=On rings whose simple modules are injective|journal=[[Journal of Algebra]]|volume=25|issue=1|pages=185–201|last1=Michler|first1=G.O.|last2=Villamayor|first2=O.E.|date=April 1973|doi=10.1016/0021-8693(73)90088-4|doi-access=free|hdl=20.500.12110/paper_00218693_v25_n1_p185_Michler|hdl-access=free}}</ref> |
A [[commutative ring]] is a V-ring [[if and only if]] it is [[Von Neumann regular ring|Von Neumann regular]].<ref>{{cite journal|title=On rings whose simple modules are injective|journal=[[Journal of Algebra]]|volume=25|issue=1|pages=185–201|last1=Michler|first1=G.O.|last2=Villamayor|first2=O.E.|date=April 1973|doi=10.1016/0021-8693(73)90088-4|doi-access=free|hdl=20.500.12110/paper_00218693_v25_n1_p185_Michler|hdl-access=free}}</ref> |
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Revision as of 21:56, 31 January 2024
In mathematics, a V-ring is a ring R such that every simple R-module is injective. The following three conditions are equivalent:[1]
- Every simple left (respectively right) R-module is injective.
- The radical of every left (respectively right) R-module is zero.
- Every left (respectively right) ideal of R is an intersection of maximal left (respectively right) ideals of R.
A commutative ring is a V-ring if and only if it is Von Neumann regular.[2]
References
- ^ Faith, Carl (1973). Algebra: Rings, modules, and categories. Springer-Verlag. ISBN 978-0387055510. Retrieved 24 October 2015.
- ^ Michler, G.O.; Villamayor, O.E. (April 1973). "On rings whose simple modules are injective". Journal of Algebra. 25 (1): 185–201. doi:10.1016/0021-8693(73)90088-4. hdl:20.500.12110/paper_00218693_v25_n1_p185_Michler.
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