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单词 SemiprimitiveRing
释义

semiprimitive ring


A ring is said to be semiprimitive if its Jacobson radicalMathworldPlanetmath is the zero idealMathworldPlanetmathPlanetmath.

Any simple ringMathworldPlanetmath is automatically semiprimitive.

A finite direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of matrix rings over division rings can be shown to be semiprimitive and both left and right ArtinianPlanetmathPlanetmath.

The Artin-Wedderburn Theorem (http://planetmath.org/WedderburnArtinTheorem) states that any semiprimitive ring which is left or right Artinian is isomorphicPlanetmathPlanetmathPlanetmath to a finite direct product of matrix rings over division rings.

Note:The semiprimitive condition is sometimes also referred to as a semisimplePlanetmathPlanetmathPlanetmath, Jacobson semisimple, or J-semisimple. Furthermore, when either of the last two names are used, the adjective ’semisimple’ is frequently intended to refer to a ring that is semiprimitive and Artinian (see the entry on semisimple ringsMathworldPlanetmath (http://planetmath.org/SemisimpleRing2)).

Titlesemiprimitive ring
Canonical nameSemiprimitiveRing
Date of creation2013-03-22 12:36:14
Last modified on2013-03-22 12:36:14
Owneryark (2760)
Last modified byyark (2760)
Numerical id20
Authoryark (2760)
Entry typeDefinition
Classificationmsc 16N20
Synonymsemisimple ring
SynonymJacobson semisimple ring
SynonymJ-semisimple ring
Synonymsemi-primitive ring
Synonymsemi-simple ring
SynonymJacobson semi-simple ring
SynonymJ-semi-simple ring
Related topicSemisimpleRing2
Related topicWedderburnArtinTheorem
Definessemiprimitivity
Definessemiprimitive
Definessemisimple
DefinesJacobson semisimple
DefinesJ-semisimple
Definessemi-primitivity
Definessemi-primitive
Definessemi-simple
DefinesJacobson semi-simple
DefinesJ-semi-simple
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更新时间:2025/5/4 23:02:45