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