Editing Semiprimitive ring

{{Ring theory sidebar}}
In [[algebra]], a '''semiprimitive ring''' or '''Jacobson semisimple ring''' or '''J-semisimple ring''' is a [[ring (mathematics)|ring]] whose [[Jacobson radical]] is [[zero ideal|zero]]. This is a type of ring more general than a [[semisimple ring]], but where [[simple module]]s still provide enough information about the ring.  Rings such as the ring of [[Integer#Algebraic properties|integers]] are semiprimitive, and an [[artinian ring|artinian]] semiprimitive ring is just a [[semisimple ring]].  Semiprimitive rings can be understood as [[subdirect product]]s of [[primitive ring]]s, which are described by the [[Jacobson density theorem]].

==Definition==
A ring is called '''semiprimitive''' or '''Jacobson semisimple''' if its Jacobson radical is the zero [[ideal (ring theory)|ideal]].

A ring is semiprimitive if and only if it has a [[faithful module|faithful]] [[semisimple module|semisimple left module]].  The semiprimitive property is left-right symmetric, and so a ring is semiprimitive if and only if it has a faithful semisimple right module.

A ring is semiprimitive if and only if it is a subdirect product of left primitive rings.

A [[commutative ring]] is semiprimitive if and only if it is a subdirect product of [[field (mathematics)|fields]], {{harv|Lam|1995|p=137}}.

A left [[artinian ring]] is semiprimitive if and only if it is [[semisimple ring|semisimple]], {{harv|Lam|2001|p=54}}. Such rings are sometimes called [[semisimple Artinian]], {{harv|Kelarev|2002|p=13}}.

==Examples==

* The ring of integers is semiprimitive, but not semisimple.
* Every primitive ring is semiprimitive.
* The [[product ring|product]] of two fields is semiprimitive but not primitive.
* Every [[von Neumann regular ring]] is semiprimitive.

[[Nathan Jacobson|Jacobson]] himself has defined a ring to be "semisimple" if and only if it is a subdirect product of [[simple ring]]s, {{harv|Jacobson|1989|p=203}}.  However, this is a stricter notion, since the [[endomorphism ring]] of a [[countably infinite]] [[dimension (vector space)|dimensional]] [[vector space]] is semiprimitive, but not a subdirect product of simple rings, {{harv|Lam|1995|p=42}}.

==References==
*{{Citation | last1=Jacobson | first1=Nathan | author1-link=Nathan Jacobson | title=Basic algebra II | publisher=W. H. Freeman | edition=2nd | isbn=978-0-7167-1933-5 | year=1989}}
*{{Citation | last1=Lam | first1=Tsit-Yuen | title=Exercises in classical ring theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Problem Books in Mathematics | isbn=978-0-387-94317-6 |mr=1323431 | year=1995}}
*{{Citation | last1=Lam | first1=Tsit-Yuen | title=A First Course in Noncommutative Rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-95325-0 | year=2001}}
* {{Citation |first1=Andrei V. |last1=Kelarev|title=Ring Constructions and Applications|year=2002|publisher=World Scientific|isbn=978-981-02-4745-4}}

[[Category:Algebraic structures]]
[[Category:Ring theory]]


{{Abstract-algebra-stub}}