Semisimple Artinian ring - Revision history

Vipul: 6 revisions

2008-05-12T16:34:42Z

6 revisions

← Older revision	Revision as of 16:34, 12 May 2008
(No difference)

Vipul: /* Weaker properties */

2008-05-06T14:14:13Z

Weaker properties

← Older revision		Revision as of 14:14, 6 May 2008
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	* [[Artinian ring]]		* [[Artinian ring]]
	* [[Zero-dimensional ring]]		* [[Zero-dimensional ring]]
	* [[~~Ring with trivial Jacobson radical~~]]: (some people use the term '''semisimple''' for such a ring)		* [[Semiprimitive ring]]: (some people use the term '''semisimple''' for such a ring)

Vipul at 14:07, 6 May 2008

2008-05-06T14:07:42Z

← Older revision		Revision as of 14:07, 6 May 2008
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	===Symbol-free definition===		===Symbol-free definition===

	A [[commutative unital ring]] is termed '''semisimple''' if it satisfies the following equivalent conditions:		A [[commutative unital ring]] is termed '''semisimple Artinian''' if it satisfies the following equivalent conditions:

	* Every module over it is [[semisimple module\|semisimple]]		* Every module over it is [[semisimple module\|semisimple]]
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	* Every short exact sequence of modules over it, splits		* Every short exact sequence of modules over it, splits
	* Its [[global dimension]] is zero		* Its [[global dimension]] is zero
			* The ring is semisimple as a module over itself
			* The ring is a direct product of finitely many fields

	==Relation with other properties==		==Relation with other properties==
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	* [[Field]]		* [[Field]]
	* [[Finite direct product of fields]]

	===Weaker properties===		===Weaker properties===

	* [[Zero-dimensional ring]]~~: {{proofat\|~~[[~~Semisimple implies zero-dimensional~~]]}}		* [[Artinian ring]]
			* [[Zero-dimensional ring]]
			* [[Ring with trivial Jacobson radical]]: (some people use the term '''semisimple''' for such a ring)

Vipul: Semisimple ring moved to Semisimple Artinian ring

2008-05-06T14:05:31Z

Semisimple ring moved to Semisimple Artinian ring

← Older revision	Revision as of 14:05, 6 May 2008
(No difference)

Vipul at 01:37, 6 May 2008

2008-05-06T01:37:53Z

← Older revision		Revision as of 01:37, 6 May 2008
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	===Symbol-free definition===		===Symbol-free definition===

	A [[commutative unital ring]] ~~(or any [[commutative ring]])~~ is termed '''semisimple''' if it satisfies the following equivalent conditions:		A [[commutative unital ring]] is termed '''semisimple''' if it satisfies the following equivalent conditions:

	* ~~The~~ [[~~Jacobson radical~~]] ~~(viz the intersection of its~~ [[~~maximal ideal~~]]~~s) is trivial~~		* Every module over it is [[semisimple module\|semisimple]]
	* It is a [[~~subdirect product~~]] of ~~fields~~		* Every module over it is [[projective module\|projective]]
	* It is a [[direct product]] ~~of fields~~		* Every module over it is [[injective module\|injective]]
			* Every short exact sequence of modules over it, splits
			* Its [[global dimension]] is zero

			==Relation with other properties==

			===Stronger properties===

			* [[Field]]
			* [[Finite direct product of fields]]

			===Weaker properties===

			* [[Zero-dimensional ring]]: {{proofat\|[[Semisimple implies zero-dimensional]]}}

Vipul at 21:06, 5 May 2008

2008-05-05T21:06:39Z

@@ Line 1: / Line 1: @@
-==Definition for commutative rings==
+{{curing property}}
 ===Symbol-free definition===
@@ Line 8: / Line 9: @@
 * It is a [[subdirect product]] of fields
 * It is a [[direct product]] of fields

Vipul: Started the page

2007-01-09T04:14:15Z

Started the page

New page

==Definition for commutative rings==

===Symbol-free definition===

A [[commutative unital ring]] (or any [[commutative ring]]) is termed '''semisimple''' if it satisfies the following equivalent conditions:

* The [[Jacobson radical]] (viz the intersection of its [[maximal ideal]]s) is trivial
* It is a [[subdirect product]] of fields
* It is a [[direct product]] of fields

==Definition for noncommutative rings==

===Symbol-free definition===

''Please check the content here''

A [[unital ring]] (or any [[ring]]) is termed '''semisimple''' if it satisfies the following equivalent conditions:

* The intersection of all its [[maximal left ideal]]s is trivial
* The intersection of all its [[maximal right ideal]]s is trivial

[[Category: Properties of noncommutative rings]]