Semiprimitive ring
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Through Algebra, Ring (mathematics) and Integer
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).
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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).