Semiprimitive ring
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In algebra, a semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. This is a type of ring more general than a semisimple ring, but where simple modules still provide enough information about the ring. Rings such as the ring of integers are semiprimitive, and an artinian semiprimitive ring is just a semisimple ring. Semiprimitive rings can be understood as subdirect products of primitive rings, 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.
A ring is semiprimitive if and only if it has a faithful 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 fields, {{harv|Lam|1995|p=137}}.
A left artinian ring is semiprimitive if and only if it is 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 of two fields is semiprimitive but not primitive.
- Every von Neumann regular ring is semiprimitive.
Jacobson himself has defined a ring to be "semisimple" if and only if it is a subdirect product of simple rings, {{harv|Jacobson|1989|p=203}}. However, this is a stricter notion, since the endomorphism ring of a countably infinite 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}}
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