Semiprimitive ring

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, (Lam 1995, p. 137).

A left artinian ring is semiprimitive if and only if it is semisimple, (Lam 2001, p. 54). Such rings are sometimes called semisimple Artinian, (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, (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, (Lam 1995, p. 42).

References

Jacobson, Nathan (1989), Basic algebra II (2nd ed.), W. H. Freeman, ISBN 978-0-7167-1933-5
Lam, Tsit-Yuen (1995), Exercises in classical ring theory, Problem Books in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94317-6, MR 1323431
Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0
Kelarev, Andrei V. (2002), Ring Constructions and Applications, World Scientific, ISBN 978-981-02-4745-4

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