Jacobson radical  Given a unital (typically noncommutative) ring
R, the Jacobson radical J(R) is defined as the set of elements
rR satisfying the following equivalent properties:    1. For
each simple left R-module M, rM=0.   2. Each maximal left ideal
of R contains r.   3. For all xR, 1rx is left invertible in R.
1. J(R) is the intersection of all maximal left ideals of R.
2. J(R) is the intersection of all maxaimal right ideals of R.
The properties required remain the same if one interchanges left
and right (modules, invertibility etc.) i.e. J(R)=J(R op).  J(R)
is a 2-sided ideal in R. The rings for which J(R)=0 are called
semiprimitive rings. In other words, for each nonzero element r
in a semiprimitive ring, by the definition, there is a simple
module left annihilated by r. Given any ring R, the quotient
R/J(R) is semiprimitive.  Revised on February 19, 2010 16:34:31
by Zoran Škoda (