Static pressure - Revision history

en>Dexbot: Bot: Fixing broken section link

2014-11-05T04:59:33Z

Bot: Fixing broken section link

← Older revision		Revision as of 06:59, 5 November 2014
Line 1:		Line 1:
	~~{{mergeto\|semisimple module\|date=April 2013}}~~		Nice to satisfy you, my name is Refugia. The favorite hobby for my kids and me is to play baseball and I'm trying to make it a occupation. My day job is a librarian. Puerto Rico is where he and his spouse live.<br><br>my site ... [http://videoworld.com/user/SMaloney home std test]
	~~{{unreferenced\|date=April 2013}}~~
	In [[ring theory]], a branch of mathematics, a '''semisimple algebra''' is an [[associative algebra\|associative]] [[artinian ring\|artinian]] algebra over a [[field (mathematics)\|field]] which has trivial [[Jacobson radical]] (only the zero element of the algebra is in the Jacobson radical). If the algebra is finite dimensional this is equivalent to ~~saying that it can be expressed as a Cartesian product of [[simple algebra\|simple subalgebras]].~~

	~~==Definition==~~
	The [[Jacobson radical]] of an algebra over a field is the ideal consisting of all elements that annihilate every simple left-module. The radical contains all nilpotent ideals, and if the algebra is finite-dimensional, ~~the radical itself is a nilpotent ideal. A finite dimensional algebra~~ is ~~then said to be ''semisimple'' if its radical contains only the zero element.~~

	~~An algebra ''A'' is called ''simple'' if it has no proper ideals and ''A''<sup>2</sup> = {''ab'' \| ''a'', ''b'' ∈ ''A''} ≠ {0}. As the terminology suggests, simple algebras are semisimple~~. The only possible ideals of a simple algebra ''A'' are ''A'' and {0}. Thus if ''A'' is not nilpotent, then ''A'' is semisimple. Because ''A''<sup>2</sup> is an ideal of ''A'' and ''A'' is simple, ''A''<sup>2</sup> = ''A''. By induction, ''A<sup>n</sup>'' = ''A'' for ~~every positive integer ''n'', i.e. ''A'' is not nilpotent.~~

	Any self-adjoint subalgebra ''A'' of ''n'' × ''n'' matrices with complex entries is semisimple. Let Rad(''A'') be the radical of ''A''. Suppose a matrix ''M'' is in Rad(''A''). Then ''MM'' lies in some nilpotent ideals of ''A'', therefore (''MM'')''<sup>k</sup>'' = 0 for some positive integer ''k''. By positive-semidefiniteness of ''MM'', this implies ''MM'' = 0. So ''M x'' is the zero vector for all ''x'', i.e. ''M'' = 0.

	~~If {''A<sub>i</sub>''} is a finite collection of simple algebras, then their Cartesian product ∏ ''A<sub>i</sub>'' is semisimple. If (''a<sub>i</sub>'') is an element of Rad(''A'')~~ and ''e''<sub>1</sub> is the multiplicative identity in ''A''<sub>1</sub> (all simple algebras possess a multiplicative identity), then (''a''<sub>1</sub>, ''a''<sub>2</sub>, ...) · (''e''<sub>1</sub>, 0, ...) = (''a''<sub>1</sub>, 0..., 0) lies in some nilpotent ideal of ∏ ''A<sub>i</sub>''. This implies, for all ''b'' in ''A''<sub>1</sub>, ''a''<sub>1</sub>''b'' is ~~nilpotent in ''A''<sub>1</sub>, i.e. ''a''<sub>1</sub> ∈ Rad(''A''<sub>1</sub>). So ''a''<sub>1</sub> = 0. Similarly, ''a<sub>i</sub>'' = 0 for all other ''i''.~~

	~~It is less apparent from the definition that the converse of the above is also true, that is, any finite dimensional semisimple algebra is isomorphic~~ to a Cartesian product of a finite number of simple algebras. The following is a semisimple algebra that appears not to be of this form. Let ''A'' be an algebra with Rad(''A'') ≠ ''A''. The quotient algebra ''B'' = ''A'' ⁄ Rad(''A'') is semisimple: If ''J'' is a nonzero nilpotent ideal in ''B'', then its preimage under the natural projection map is a nilpotent ideal in ''A'' which is strictly larger than Rad(''A''), a contradiction.

	~~==Characterization==~~
	~~Let ''A'' be a finite dimensional semisimple algebra,~~ and

	~~:<math>\{0\} = J_0 \subset \cdots \subset J_n \subset A</math>~~

	~~be a [[composition series]] of ''A'', then ''A'~~' ~~is isomorphic~~ to ~~the following Cartesian product:~~

	~~:<math>A \simeq J_1 \times J_2/J_1 \times J_3/J_2 \times ... \times J_n/ J_{n-1} \times A / J_n </math>~~

	~~where each~~

	~~:<math>J_{i+1}/J_i \,</math>~~

	is a ~~simple algebra.~~

	~~The proof can be sketched as follows~~. ~~First, invoking the assumption that ''A'' is semisimple, one can show that the ''J''<sub>1</sub>~~ is a ~~simple algebra (therefore unital)~~. ~~So ''J''<sub>1</sub>~~ is ~~a unital subalgebra~~ and ~~an ideal of ''J''<sub>2</sub>~~. ~~Therefore one can decompose~~

	~~:<math>J_2 \simeq J_1 \times J_2/J_1 .</math>~~

	~~By maximality of ''J''<sub>1</sub> as an ideal in ''J''<sub>2~~<~~/sub~~> ~~and also the semisimplicity of ''A'', the algebra~~

	:<~~math~~>~~J_2/J_1 \,</math>~~

	~~is simple~~. ~~Proceed by induction in similar fashion proves the claim~~. ~~For example, ''J''<sub>3</sub> is the Cartesian product of simple algebras~~

	~~:<math>J_3 \simeq J_2 \times J_3 / J_2 \simeq J_1 \times J_2/J_1 \times J_3 / J_2.</math>~~

	~~The above result can be restated in a different way~~. For a semisimple algebra ''A'' = ''A''<sub>1</sub> ×...× ''A<sub>n</sub>'' expressed in terms of its simple factors, consider the units ''e<sub>i</sub>'' ∈ ''A<sub>i</sub>''. The elements ''E<sub>i</sub>'' = (0,...,''e<sub>i</sub>'',...,0) are [[idempotent element]]s in ''A'' and they lie in the center of ''A''. Furthermore, ''E<sub>i</sub> A'' = ''A<sub>i</sub>'', ''E<sub>i</sub>E<sub>j</sub>'' = 0 for ''i'' ≠ ''j'', and Σ ''E<sub>i</sub>'' = 1, the multiplicative identity in ''A''.

	~~Therefore, for every semisimple algebra ''A'', there exists idempotents {''E<sub>i</sub>''} in the center of ''A'', such that~~

	~~#''E<sub>i</sub>E<sub>j</sub>'' = 0 for ''i'' ≠ ''j'' (such a set of idempotents is called ''[[Idempotent_element#Types_of_ring_idempotents\|central orthogonal]]''),~~
	~~#Σ ''E<sub>i</sub>'' = 1,~~
	~~#''A'' is isomorphic to the Cartesian product of simple algebras ''E''<sub>1</sub> ''A'' ×...× ''E<sub>n</sub> A''.~~

	~~==Classification==~~
	The [[Artin–Wedderburn theorem]] completely classifies semisimple algebras: they are isomorphic to a product <math> \prod M_{n_i}(D_i) </math> where the <math> n_i </math> are some integers, the <math> D_i </math> are [[division ring]]s, and <math> M_{n_i}(D_i) </math> means the ring of <math> n_i \times n_i </math> matrices over <math> D_i</math>. This product is unique up to permutation of the factors.

	~~==References==~~
	[http://~~www.encyclopediaofmath~~.~~org~~/~~index.php~~/~~Semi-simple_algebra Springer Encyclopedia of Mathematics]~~


	~~{{DEFAULTSORT:Semisimple Algebra}}~~
	~~[[Category:Algebras]~~]

en>Dolphin51: Rv g-f edit by 173.89.15.69 (talk). That is not what the sources say, and it contradicts what is said elsewhere in the article. Discussed on the IP's Talk page.

2014-01-20T03:16:33Z

Rv g-f edit by 173.89.15.69 (talk). That is not what the sources say, and it contradicts what is said elsewhere in the article. Discussed on the IP's Talk page.

← Older revision		Revision as of 05:16, 20 January 2014
Line 1:		Line 1:
	~~Myrtle Benny~~ is ~~how~~ ~~std home test I~~'m called and ~~I really feel comfortable when people use~~ the ~~full name~~. ~~For many years home std test he~~'~~s been working as~~ a ~~meter reader~~ and it'~~s something he [http:~~//~~www~~.~~revleft~~.~~com~~/vb/~~member~~.~~php?u~~=~~160656 www~~.~~revleft~~.~~com] really enjoy~~. ~~Playing baseball std testing at home~~ is the ~~pastime he will by no means stop performing~~. [~~http~~://~~www~~.~~ign~~.~~com~~/~~boards~~/~~threads~~/~~list-~~of~~-celebrities-with-herpes~~.~~452635999~~/ ~~California]~~ is ~~our birth location~~.<br><br>~~Feel [http~~://~~product-dev~~.~~younetco~~.~~com~~/~~xuannth~~/~~fox_ios1~~/~~index~~.~~php?do~~=/~~profile-1539~~/~~info~~/ ~~at home std test~~] ~~free~~ to ~~surf to my blog post~~; [~~https~~://~~alphacomms~~.~~zendesk~~.~~com~~/~~entries~~/~~53455304-Curing-Your-Candida-Albicans-How-To-Make-It-Happen-Easily alphacomms~~.~~zendesk~~.~~com~~]		{{mergeto\|semisimple module\|date=April 2013}}
			{{unreferenced\|date=April 2013}}
			In [[ring theory]], a branch of mathematics, a '''semisimple algebra''' is an [[associative algebra\|associative]] [[artinian ring\|artinian]] algebra over a [[field (mathematics)\|field]] which has trivial [[Jacobson radical]] (only the zero element of the algebra is in the Jacobson radical). If the algebra is finite dimensional this is equivalent to saying that it can be expressed as a Cartesian product of [[simple algebra\|simple subalgebras]].

			==Definition==
			The [[Jacobson radical]] of an algebra over a field is the ideal consisting of all elements that annihilate every simple left-module. The radical contains all nilpotent ideals, and if the algebra is finite-dimensional, the radical itself is a nilpotent ideal. A finite dimensional algebra is then said to be ''semisimple'' if its radical contains only the zero element.

			An algebra ''A'' is called ''simple'' if it has no proper ideals and ''A''<sup>2</sup> = {''ab'' \| ''a'', ''b'' ∈ ''A''} ≠ {0}. As the terminology suggests, simple algebras are semisimple. The only possible ideals of a simple algebra ''A'' are ''A'' and {0}. Thus if ''A'' is not nilpotent, then ''A'' is semisimple. Because ''A''<sup>2</sup> is an ideal of ''A'' and ''A'' is simple, ''A''<sup>2</sup> = ''A''. By induction, ''A<sup>n</sup>'' = ''A'' for every positive integer ''n'', i.e. ''A'' is not nilpotent.

			Any self-adjoint subalgebra ''A'' of ''n'' × ''n'' matrices with complex entries is semisimple. Let Rad(''A'') be the radical of ''A''. Suppose a matrix ''M'' is in Rad(''A''). Then ''MM'' lies in some nilpotent ideals of ''A'', therefore (''MM'')''<sup>k</sup>'' = 0 for some positive integer ''k''. By positive-semidefiniteness of ''MM'', this implies ''MM'' = 0. So ''M x'' is the zero vector for all ''x'', i.e. ''M'' = 0.

			If {''A<sub>i</sub>''} is a finite collection of simple algebras, then their Cartesian product ∏ ''A<sub>i</sub>'' is semisimple. If (''a<sub>i</sub>'') is an element of Rad(''A'') and ''e''<sub>1</sub> is the multiplicative identity in ''A''<sub>1</sub> (all simple algebras possess a multiplicative identity), then (''a''<sub>1</sub>, ''a''<sub>2</sub>, ...) · (''e''<sub>1</sub>, 0, ...) = (''a''<sub>1</sub>, 0..., 0) lies in some nilpotent ideal of ∏ ''A<sub>i</sub>''. This implies, for all ''b'' in ''A''<sub>1</sub>, ''a''<sub>1</sub>''b'' is nilpotent in ''A''<sub>1</sub>, i.e. ''a''<sub>1</sub> ∈ Rad(''A''<sub>1</sub>). So ''a''<sub>1</sub> = 0. Similarly, ''a<sub>i</sub>'' = 0 for all other ''i''.

			It is less apparent from the definition that the converse of the above is also true, that is, any finite dimensional semisimple algebra is isomorphic to a Cartesian product of a finite number of simple algebras. The following is a semisimple algebra that appears not to be of this form. Let ''A'' be an algebra with Rad(''A'') ≠ ''A''. The quotient algebra ''B'' = ''A'' ⁄ Rad(''A'') is semisimple: If ''J'' is a nonzero nilpotent ideal in ''B'', then its preimage under the natural projection map is a nilpotent ideal in ''A'' which is strictly larger than Rad(''A''), a contradiction.

			==Characterization==
			Let ''A'' be a finite dimensional semisimple algebra, and

			:<math>\{0\} = J_0 \subset \cdots \subset J_n \subset A</math>

			be a [[composition series]] of ''A'', then ''A'' is isomorphic to the following Cartesian product:

			:<math>A \simeq J_1 \times J_2/J_1 \times J_3/J_2 \times ... \times J_n/ J_{n-1} \times A / J_n </math>

			where each

			:<math>J_{i+1}/J_i \,</math>

			is a simple algebra.

			The proof can be sketched as follows. First, invoking the assumption that ''A'' is semisimple, one can show that the ''J''<sub>1</sub> is a simple algebra (therefore unital). So ''J''<sub>1</sub> is a unital subalgebra and an ideal of ''J''<sub>2</sub>. Therefore one can decompose

			:<math>J_2 \simeq J_1 \times J_2/J_1 .</math>

			By maximality of ''J''<sub>1</sub> as an ideal in ''J''<sub>2</sub> and also the semisimplicity of ''A'', the algebra

			:<math>J_2/J_1 \,</math>

			is simple. Proceed by induction in similar fashion proves the claim. For example, ''J''<sub>3</sub> is the Cartesian product of simple algebras

			:<math>J_3 \simeq J_2 \times J_3 / J_2 \simeq J_1 \times J_2/J_1 \times J_3 / J_2.</math>

			The above result can be restated in a different way. For a semisimple algebra ''A'' = ''A''<sub>1</sub> ×...× ''A<sub>n</sub>'' expressed in terms of its simple factors, consider the units ''e<sub>i</sub>'' ∈ ''A<sub>i</sub>''. The elements ''E<sub>i</sub>'' = (0,...,''e<sub>i</sub>'',...,0) are [[idempotent element]]s in ''A'' and they lie in the center of ''A''. Furthermore, ''E<sub>i</sub> A'' = ''A<sub>i</sub>'', ''E<sub>i</sub>E<sub>j</sub>'' = 0 for ''i'' ≠ ''j'', and Σ ''E<sub>i</sub>'' = 1, the multiplicative identity in ''A''.

			Therefore, for every semisimple algebra ''A'', there exists idempotents {''E<sub>i</sub>''} in the center of ''A'', such that

			#''E<sub>i</sub>E<sub>j</sub>'' = 0 for ''i'' ≠ ''j'' (such a set of idempotents is called ''[[Idempotent_element#Types_of_ring_idempotents\|central orthogonal]]''),
			#Σ ''E<sub>i</sub>'' = 1,
			#''A'' is isomorphic to the Cartesian product of simple algebras ''E''<sub>1</sub> ''A'' ×...× ''E<sub>n</sub> A''.

			==Classification==
			The [[Artin–Wedderburn theorem]] completely classifies semisimple algebras: they are isomorphic to a product <math> \prod M_{n_i}(D_i) </math> where the <math> n_i </math> are some integers, the <math> D_i </math> are [[division ring]]s, and <math> M_{n_i}(D_i) </math> means the ring of <math> n_i \times n_i </math> matrices over <math> D_i</math>. This product is unique up to permutation of the factors.

			==References==
			[http://www.encyclopediaofmath.org/index.php/Semi-simple_algebra Springer Encyclopedia of Mathematics]


			{{DEFAULTSORT:Semisimple Algebra}}
			[[Category:Algebras]]

94.192.234.132: /* Static pressure in fluid statics */

2012-08-03T22:16:21Z

Static pressure in fluid statics

New page

Myrtle Benny is how std home test I'm called and I really feel comfortable when people use the full name. For many years home std test he's been working as a meter reader and it's something he [http://www.revleft.com/vb/member.php?u=160656 www.revleft.com] really enjoy. Playing baseball std testing at home is the pastime he will by no means stop performing. [http://www.ign.com/boards/threads/list-of-celebrities-with-herpes.452635999/ California] is our birth location.<br><br>Feel [http://product-dev.younetco.com/xuannth/fox_ios1/index.php?do=/profile-1539/info/ at home std test] free to surf to my blog post; [https://alphacomms.zendesk.com/entries/53455304-Curing-Your-Candida-Albicans-How-To-Make-It-Happen-Easily alphacomms.zendesk.com]