en>Luismanu at 06:56, 6 April 2014

2014-04-06T06:56:22Z

← Older revision		Revision as of 08:56, 6 April 2014
Line 1:		Line 1:
	~~{{about\|perfect rings as introduced by Hyman Bass\|perfect rings of characteristic p generalizing perfect fields\|perfect field}}~~		My name's Regena Skuthorp but everybody calls me Regena. I'm from Great Britain. I'm studying at the high school (2nd year) and I play the Lute for 4 years. Usually I choose music from the famous films :D. <br>I have two brothers. I like Golfing, watching movies and Fishing.<br><br>my website :: [http://troptiontrading.com/2014/10/19/%d0%94%d0%b8%d0%b5%d1%82%d1%8b-%d0%ba%d0%b0%d0%ba-%d0%bf%d0%be%d1%85%d1%83%d0%b4%d0%b5%d1%82%d1%8c-%d0%b1%d1%8b%d1%81%d1%82%d1%80%d0%b5%d0%b5-%d0%b8-%d1%83%d0%bf%d1%80%d0%b0%d0%b6%d0%bd%d0%b5%d0%bd/ меню японской диеты]

	~~{{Merge from \|semiperfect ring \|date=March 2011}}~~

	~~In the area of [[abstract algebra]] known as [[ring theory]], a '''left perfect ring''~~' ~~is a type of ring in which all left [[module (algebra)\|modules]] have [[projective cover]]~~s~~. The right case is defined by analogy, and the condition is not left-right symmetric, that is, there exist rings which are perfect on one side~~ but ~~not the other~~. ~~Perfect rings were introduced in {{harv\|Bass\|1960}}.~~

	~~==Definitions==~~
	~~The following equivalent definitions of a left perfect ring~~ '~~'R'' are found in {{harv\|Anderson,Fuller\|1992, p~~.~~315}}:~~
	* Every left '~~'R'' module has a projective cover.~~
	* ''R''/J(~~''R''~~) ~~is [[semisimple module\|semisimple]]~~ and ~~J(''R'') is '''left T-nilpotent''' (that is,~~ for ~~every infinite sequence of elements of J(''R'') there is an ''n'' such that the product of first ''n'' terms are zero), where J(''R'') is the [[Jacobson radical]] of ''R''~~.
	* ('''Bass' Theorem P''') ''R'' satisfies the [[descending chain condition]] on principal right ideals. (There is no mistake, this condition on ''right'' principal ideals is equivalent to the ~~ring being ''left'' perfect.)~~
	* Every [[flat module\|flat]] left ''R''-module is [[projective module\|projective]].
	* ''R''/J(''R'') is semisimple and every non-zero left ''R'' module contains a [[maximal submodule]].
	* ''R'' contains no infinite orthogonal set of [[idempotent element\|idempotent]]s, and every non-zero right ''R'' module contains a minimal submodule.

	~~==Examples==~~
	* Right or left [[Artinian ring]]s, and [[Hopkins–Levitzki theorem\|semiprimary ring]]s are known to be right-and-left perfect.
	* The following is an example (due to Bass) of a [[local ring]] which is right but not left perfect. Let ''F'' be a field, and consider a certain ring of [[matrix (mathematics)#Infinite matrices\|infinite matrices]] over ''F''.
	:~~Take the set of infinite matrices with entries indexed by ℕ× ℕ, and which only have finitely many nonzero entries above the diagonal, and denote this set by ''J''~~. ~~Also take the matrix~~ <~~math~~>I~~\,</math> with all 1's on the diagonal~~, and ~~form the set~~
	:<~~math~~>~~R=\{f\cdot I+j\mid f\in F, j\in J \}\,~~<~~/math~~>
	:It can be shown that ''R'' is a ring with identity, whose [[Jacobson radical]] is ''J''. Furthermore ''R''/''J'' is a field, so that ''R'' is local, and ''R'' is right but not left perfect. {{harv\|Lam\|2001, p.345-346}}

	~~==Properties==~~
	~~For a left perfect ring ''R''~~:
	* From the equivalences above, every left ''R'' module has a maximal submodule and a projective cover, and the flat left ''R'' modules coincide with the projective left modules.
	* ''R'' is a [[semiperfect ring]], since one of the characterizations of semiperfect rings is: ~~"All [[finitely generated module\|finitely generated]] left ''R'' modules have projective covers."~~
	* An analogue of the [[~~Injective module#Baer's criterion\|Baer's criterion]] holds for projective modules. {{Citation needed\|date=July 2011}}~~

	~~== References ==~~
	*{{Citation\|last = Anderson\|first = Frank W\|coauthors = Fuller, Kent R\|title = Rings and Categories of Modules\|publisher = Springer\|year = 1992\|isbn = 0-387-97845-3\|url = http://~~books.google~~.com/~~?id=PswhrD_wUIkC \| pages=312–322}}~~
	* {{Citation \| last1=Bass \| first1=Hyman \| title=Finitistic dimension and a homological generalization of semi-primary rings \| doi=10~~.2307~~/~~1993568 \| jstor=1993568 \| mr=0157984 \| year=1960 \| journal=[[Transactions of the American Mathematical Society]] \| issn=0002~~-~~9947 \| volume=95 \| issue=3 \| pages=466–488}}~~
	*{{citation \|author=Lam, T. Y. \|title=A first course in noncommutative rings \|series=Graduate Texts in Mathematics \|volume=131 \|edition=2 \|publisher=Springer-~~Verlag \|place=New York \|year=2001 \|pages=xx+385 \|isbn=0~~-~~387~~-~~95183~~-~~0 \|mr=1838439 }}~~

	~~[[Category:Ring theory]~~]

en>Yobot: WP:CHECKWIKI error fixes + general fixes, References after punctuation per WP:REFPUNC and WP:PAIC using AWB (7510)

2010-12-27T08:32:23Z

WP:CHECKWIKI error fixes + general fixes, References after punctuation per WP:REFPUNC and WP:PAIC using AWB (7510)

New page

{{about|perfect rings as introduced by Hyman Bass|perfect rings of characteristic p generalizing perfect fields|perfect field}}

{{Merge from |semiperfect ring |date=March 2011}}

In the area of [[abstract algebra]] known as [[ring theory]], a '''left perfect ring''' is a type of ring in which all left [[module (algebra)|modules]] have [[projective cover]]s. The right case is defined by analogy, and the condition is not left-right symmetric, that is, there exist rings which are perfect on one side but not the other. Perfect rings were introduced in {{harv|Bass|1960}}.

==Definitions==
The following equivalent definitions of a left perfect ring ''R'' are found in {{harv|Anderson,Fuller|1992, p.315}}:
* Every left ''R'' module has a projective cover.
* ''R''/J(''R'') is [[semisimple module|semisimple]] and J(''R'') is '''left T-nilpotent''' (that is, for every infinite sequence of elements of J(''R'') there is an ''n'' such that the product of first ''n'' terms are zero), where J(''R'') is the [[Jacobson radical]] of ''R''.
* ('''Bass' Theorem P''') ''R'' satisfies the [[descending chain condition]] on principal right ideals. (There is no mistake, this condition on ''right'' principal ideals is equivalent to the ring being ''left'' perfect.)
* Every [[flat module|flat]] left ''R''-module is [[projective module|projective]].
* ''R''/J(''R'') is semisimple and every non-zero left ''R'' module contains a [[maximal submodule]].
* ''R'' contains no infinite orthogonal set of [[idempotent element|idempotent]]s, and every non-zero right ''R'' module contains a minimal submodule.

==Examples==
* Right or left [[Artinian ring]]s, and [[Hopkins–Levitzki theorem|semiprimary ring]]s are known to be right-and-left perfect.
* The following is an example (due to Bass) of a [[local ring]] which is right but not left perfect. Let ''F'' be a field, and consider a certain ring of [[matrix (mathematics)#Infinite matrices|infinite matrices]] over ''F''.
:Take the set of infinite matrices with entries indexed by ℕ× ℕ, and which only have finitely many nonzero entries above the diagonal, and denote this set by ''J''. Also take the matrix <math>I\,</math> with all 1's on the diagonal, and form the set
:<math>R=\{f\cdot I+j\mid f\in F, j\in J \}\,</math>
:It can be shown that ''R'' is a ring with identity, whose [[Jacobson radical]] is ''J''. Furthermore ''R''/''J'' is a field, so that ''R'' is local, and ''R'' is right but not left perfect. {{harv|Lam|2001, p.345-346}}

==Properties==
For a left perfect ring ''R'':
* From the equivalences above, every left ''R'' module has a maximal submodule and a projective cover, and the flat left ''R'' modules coincide with the projective left modules.
* ''R'' is a [[semiperfect ring]], since one of the characterizations of semiperfect rings is: "All [[finitely generated module|finitely generated]] left ''R'' modules have projective covers."
* An analogue of the [[Injective module#Baer's criterion|Baer's criterion]] holds for projective modules. {{Citation needed|date=July 2011}}

== References ==
*{{Citation|last = Anderson|first = Frank W|coauthors = Fuller, Kent R|title = Rings and Categories of Modules|publisher = Springer|year = 1992|isbn = 0-387-97845-3|url = http://books.google.com/?id=PswhrD_wUIkC | pages=312–322}}
* {{Citation | last1=Bass | first1=Hyman | title=Finitistic dimension and a homological generalization of semi-primary rings | doi=10.2307/1993568 | jstor=1993568 | mr=0157984 | year=1960 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=95 | issue=3 | pages=466–488}}
*{{citation |author=Lam, T. Y. |title=A first course in noncommutative rings |series=Graduate Texts in Mathematics |volume=131 |edition=2 |publisher=Springer-Verlag |place=New York |year=2001 |pages=xx+385 |isbn=0-387-95183-0 |mr=1838439 }}

[[Category:Ring theory]]

Hamming scheme - Revision history

en>Luismanu at 06:56, 6 April 2014

en>Yobot: WP:CHECKWIKI error fixes + general fixes, References after punctuation per WP:REFPUNC and WP:PAIC using AWB (7510)