# 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