en>Rhombus: The link to the standard longitude is very confusing.

2013-06-03T11:51:57Z

The link to the standard longitude is very confusing.

← Older revision		Revision as of 13:51, 3 June 2013
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			In [[mathematics]], a '''semi-local ring''' is a [[ring (mathematics)\|ring]] for which ''R''/J(''R'') is a [[semisimple ring]], where J(''R'') is the [[Jacobson radical]] of ''R''. {{harv\|Lam\|2001, §20}}{{harv\|Mikhalev\|2002, C.7}}

			The above definition implies ''R'' has a finite number of maximal right ideals (and finite number of maximal left ideals). When ''R'' is a [[commutative ring]], the converse implication is also true, and so the definition of semi-local for commutative rings is often taken to be "having finitely many [[maximal ideal]]s".

			Some literature refers to a commutative semi-local ring in general as a
			''quasi-semi-local ring'', using semi-local ring to refer to a [[Noetherian ring]] with finitely many maximal ideals.

			A semi-local ring is thus more general than a [[local ring]], which has only one maximal (right/left/two-sided) ideal.

			== Examples ==
			* Any right or left [[Artinian ring]], any [[serial ring]], and any [[semiperfect ring]] is semi-local.
			* The quotient <math>\mathbb{Z}/m\mathbb{Z}</math> is a semi-local ring. In particular, if <math>m</math> is a prime power, then <math>\mathbb{Z}/m\mathbb{Z}</math> is a local ring.
			* A finite direct sum of fields <math>\bigoplus_{i=1}^n{F_i}</math> is a semi-local ring.
			* In the case of commutative rings with unity, this example is prototypical in the following sense: the [[Chinese remainder theorem]] shows that for a semi-local commutative ring ''R'' with unit and maximal ideals ''m<sub>1</sub>, ..., m<sub>n</sub>''
			:<math>R/\bigcap_{i=1}^n m_i\cong\bigoplus_{i=1}^n R/m_i\,</math>.
			:(The map is the natural projection). The right hand side is a direct sum of fields. Here we note that ∩<sub>i</sub> m<sub>i</sub>=J(''R''), and we see that ''R''/J(''R'') is indeed a semisimple ring.
			* The [[classical ring of quotients]] for any commutative Noetherian ring is a semilocal ring.
			* The [[endomorphism ring]] of an [[Artinian module]] is a semilocal ring.
			* Semi-local rings occur for example in [[algebraic geometry]] when a (commutative) ring ''R'' is [[Localization of a ring\|localized]] with respect to the multiplicatively closed subset ''S = ∩ (R \ p<sub>i</sub>)'', where the ''p<sub>i</sub>'' are finitely many [[prime ideal]]s.

			==Textbooks==
			*{{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
			\|date=2001
			\|chapter=7
			\|pages=xx+385
			\|isbn=0-387-95183-0
			\|mr=1838439 (2002c:16001)}
			}}
			*{{citation
			\|title=The concise handbook of algebra
			\|editor1=Mikhalev, Alexander V.
			\|editor2=Pilz, Günter F.
			\|publisher=Kluwer Academic Publishers
			\|place=Dordrecht
			\|date=2002
			\|pages=xvi+618
			\|isbn=0-7923-7072-4
			\|mr=1966155 (2004c:00001)
			}}

			[[Category:Ring theory]]
			[[Category:Localization (mathematics)]]


			{{Abstract-algebra-stub}}

en>Teapeat: celestial mechanics

2012-06-27T19:30:26Z

celestial mechanics

New page

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True longitude - Revision history

en>Rhombus: The link to the standard longitude is very confusing.

en>Teapeat: celestial mechanics