nLab
Jacobson radical

Given a unital (typically noncommutative) ring RR, the Jacobson radical J(R)J(R) is defined as the set of elements rRr\in R satisfying the following equivalent properties:

  1. For each simple left RR-module MM, rM=0r M =0.
  2. Each maximal left ideal of RR contains rr.
  3. For all xRx\in R, 1rx1 - r x is left invertible in RR.

Alternatively,

  1. J(R)J(R) is the intersection of all maximal left ideals of RR.
  2. J(R)J(R) is the intersection of all maxaimal right ideals of RR.

The properties required remain the same if one interchanges left and right (modules, invertibility etc.) i.e. J(R)=J(R op)J(R)=J(R^{op}).

J(R)J(R) is a 22-sided ideal in RR. The rings for which J(R)=0J(R)=0 are called semiprimitive rings. In other words, for each nonzero element rr in a semiprimitive ring, by the definition, there is a simple module left annihilated by rr. Given any ring RR, the quotient R/J(R)R/J(R) is semiprimitive.

Revised on February 19, 2010 16:34:31 by Zoran Škoda (161.53.130.104)