en>Colonies Chris: /* Generalization */sp, date & link fixes; unlinking common words, replaced: Kac-Moody → Kac–Moody (4) using AWB

2014-03-11T22:44:56Z

Generalization: sp, date & link fixes; unlinking common words, replaced: Kac-Moody → Kac–Moody (4) using AWB

← Older revision		Revision as of 00:44, 12 March 2014
Line 1:		Line 1:
	~~{{Merge to \|perfect ring \|date=March 2011}}~~		Ed is what people contact me and my spouse doesn't like it at all. To climb is something she would by no means give up. For a while I've been in Mississippi but now I'm considering other choices. I am currently a journey agent.<br><br>Have a look at my weblog :: best psychic, [http://formalarmour.com/index.php?do=/profile-26947/info/ http://formalarmour.com],


	~~In [[abstract algebra]], a '''semiperfect ring'''~~ is a [[ring (mathematics)\|ring]] over which every [[finitely generated module\|finitely generated]] left [[module (mathematics)\|module]] has a [[projective cover]]. This property is left-right symmetric.

	~~== Definition ==~~

	~~Let ''R'~~' ~~be ring~~. ~~Then ''R'' is '''semiperfect''' if any of the following equivalent conditions hold:~~

	* ''R''/J(''R'') is [[semisimple module\|semisimple]] and [[idempotent element\|idempotent]]s lift modulo J(''R''), where J(''R'') is ~~the [[Jacobson radical]] of ''R''~~.
	* ''R'' has a ~~complete orthogonal set~~ ''~~e''<sub>1</sub>,~~ ..~~., ''e''~~<~~sub~~>~~''n''</sub> of idempotents with each ''e''~~<~~sub>''i''</sub> ''R e''<sub>''i''</sub~~> a ~~[[local ring]].~~
	* Every [[simple module\|simple]] left (right) [[module (mathematics)\|''R''-module]] has a [[projective cover]].
	* Every [[finitely generated module\|finitely generated]] left (right) ''R''-module has a projective cover.
	* The category of finitely generated projective <math>R</math>-modules is [[Krull-Schmidt_category\|Krull-Schmidt]].

	~~== Examples ==~~

	~~Examples of '''semiperfect rings''' include~~:

	* Left (right) [[perfect ring]]s.
	* [[Local ring]]s.
	* Left (right) [[Artinian ring]]s.
	* [[dimension (vector space)\|Finite dimensional]] [[algebra over a field\|''k''-algebras]].

	~~== Properties ==~~

	~~Since a ring ''R'' is semiperfect iff every [[simple module\|simple]] left [[module (mathematics)\|''R''-module]] has a projective cover~~, ~~every ring~~ [~~[Morita equivalence\|Morita equivalent]] to a semiperfect ring is also semiperfect.~~

	~~== References ==~~

	*{{cite book\|last = Anderson\|first = Frank Wylie\|coauthors = Fuller, Kent R\|title = Rings and Categories of Modules\|publisher = Springer\|date = 1992\|isbn = 0-387-97845-3\|url = http://~~books.google~~.com/~~books~~?id=~~PswhrD_wUIkC\|accessdate = 2007~~-~~03-27}}~~

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

en>Yobot: WP:CHECKWIKI error 18 fixes + general fixes (BRFA 15) using AWB (7832)

2011-09-09T10:52:35Z

WP:CHECKWIKI error 18 fixes + general fixes (BRFA 15) using AWB (7832)

New page

{{Merge to |perfect ring |date=March 2011}}

In [[abstract algebra]], a '''semiperfect ring''' is a [[ring (mathematics)|ring]] over which every [[finitely generated module|finitely generated]] left [[module (mathematics)|module]] has a [[projective cover]]. This property is left-right symmetric.

== Definition ==

Let ''R'' be ring. Then ''R'' is '''semiperfect''' if any of the following equivalent conditions hold:

* ''R''/J(''R'') is [[semisimple module|semisimple]] and [[idempotent element|idempotent]]s lift modulo J(''R''), where J(''R'') is the [[Jacobson radical]] of ''R''.
* ''R'' has a complete orthogonal set ''e''1, ..., ''e''''n'' of idempotents with each ''e''''i'' ''R e''''i'' a [[local ring]].
* Every [[simple module|simple]] left (right) [[module (mathematics)|''R''-module]] has a [[projective cover]].
* Every [[finitely generated module|finitely generated]] left (right) ''R''-module has a projective cover.
* The category of finitely generated projective <math>R</math>-modules is [[Krull-Schmidt_category|Krull-Schmidt]].

== Examples ==

Examples of '''semiperfect rings''' include:

* Left (right) [[perfect ring]]s.
* [[Local ring]]s.
* Left (right) [[Artinian ring]]s.
* [[dimension (vector space)|Finite dimensional]] [[algebra over a field|''k''-algebras]].

== Properties ==

Since a ring ''R'' is semiperfect iff every [[simple module|simple]] left [[module (mathematics)|''R''-module]] has a projective cover, every ring [[Morita equivalence|Morita equivalent]] to a semiperfect ring is also semiperfect.

== References ==

*{{cite book|last = Anderson|first = Frank Wylie|coauthors = Fuller, Kent R|title = Rings and Categories of Modules|publisher = Springer|date = 1992|isbn = 0-387-97845-3|url = http://books.google.com/books?id=PswhrD_wUIkC|accessdate = 2007-03-27}}

[[Category:Ring theory]]
[[Category:Module theory]]

Algebraic character - Revision history

en>Colonies Chris: /* Generalization */sp, date & link fixes; unlinking common words, replaced: Kac-Moody → Kac–Moody (4) using AWB

en>Yobot: WP:CHECKWIKI error 18 fixes + general fixes (BRFA 15) using AWB (7832)