Stably finite ring: Difference between revisions
Content deleted Content added
Fixed URL temporarily. |
Filled in 1 bare reference(s) with reFill 2 |
||
| Line 1: | Line 1: | ||
In [[mathematics]], particularly in [[abstract algebra]], a [[ring (mathematics)|ring]] ''R'' is said to be '''stably finite''' (or '''weakly finite''') if, for all [[square matrices]] ''A'' and ''B'' of the same size with entries in ''R'', ''AB'' = 1 implies ''BA'' = 1.<ref>https://books.google.com/books?id=bOElBQAAQBAJ&newbks=1&newbks_redir=0&printsec=frontcover#v=onepage&q&f=false</ref> This is a stronger property for a ring than having the [[invariant basis number]] (IBN) property. Namely, any [[trivial ring|nontrivial]]<ref group="notes">A trivial ring is stably finite but doesn't have IBN.</ref> stably finite ring has IBN. [[Commutative ring]]s, [[noetherian ring]]s and [[artinian ring]]s are stably finite. [[Subring]]s of stably finite rings and [[matrix ring]]s over stably finite rings are stably finite. A ring satisfying [[Klein's nilpotence condition]] is stably finite.<ref>{{Cite web|url=https://www.google.com/books/edition/Skew_Fields/u-4ADgUgpSMC?hl=en&gbpv=1&dq=%22klein%27s+nilpotence+condition%22&pg=PA23&printsec=frontcover|title=Skew Fields: Theory of General Division Rings|first=Paul Moritz|last=Cohn|date=July 28, 1995|publisher=Cambridge University Press|via=Google Books}}</ref> |
In [[mathematics]], particularly in [[abstract algebra]], a [[ring (mathematics)|ring]] ''R'' is said to be '''stably finite''' (or '''weakly finite''') if, for all [[square matrices]] ''A'' and ''B'' of the same size with entries in ''R'', ''AB'' = 1 implies ''BA'' = 1.<ref>{{Cite web|url=https://books.google.com/books?id=bOElBQAAQBAJ&newbks=1&newbks_redir=0&printsec=frontcover#v=onepage&q&f=false|title=Basic Algebra: Groups, Rings and Fields|first=P. M.|last=Cohn|date=December 6, 2012|publisher=Springer Science & Business Media|via=Google Books}}</ref> This is a stronger property for a ring than having the [[invariant basis number]] (IBN) property. Namely, any [[trivial ring|nontrivial]]<ref group="notes">A trivial ring is stably finite but doesn't have IBN.</ref> stably finite ring has IBN. [[Commutative ring]]s, [[noetherian ring]]s and [[artinian ring]]s are stably finite. [[Subring]]s of stably finite rings and [[matrix ring]]s over stably finite rings are stably finite. A ring satisfying [[Klein's nilpotence condition]] is stably finite.<ref>{{Cite web|url=https://www.google.com/books/edition/Skew_Fields/u-4ADgUgpSMC?hl=en&gbpv=1&dq=%22klein%27s+nilpotence+condition%22&pg=PA23&printsec=frontcover|title=Skew Fields: Theory of General Division Rings|first=Paul Moritz|last=Cohn|date=July 28, 1995|publisher=Cambridge University Press|via=Google Books}}</ref> |
||
==Notes== |
==Notes== |
||
Revision as of 21:45, 18 October 2023
In mathematics, particularly in abstract algebra, a ring R is said to be stably finite (or weakly finite) if, for all square matrices A and B of the same size with entries in R, AB = 1 implies BA = 1.[1] This is a stronger property for a ring than having the invariant basis number (IBN) property. Namely, any nontrivial[notes 1] stably finite ring has IBN. Commutative rings, noetherian rings and artinian rings are stably finite. Subrings of stably finite rings and matrix rings over stably finite rings are stably finite. A ring satisfying Klein's nilpotence condition is stably finite.[2]
Notes
- ^ A trivial ring is stably finite but doesn't have IBN.
References
- ^ Cohn, P. M. (December 6, 2012). "Basic Algebra: Groups, Rings and Fields". Springer Science & Business Media – via Google Books.
- ^ Cohn, Paul Moritz (July 28, 1995). "Skew Fields: Theory of General Division Rings". Cambridge University Press – via Google Books.
 | This abstract algebra-related article is a stub. You can help Wikipedia by expanding it. |