Given a unital (typically noncommutative) ring , the Jacobson radical is defined as the set of elements satisfying the following equivalent properties:
The properties required remain the same if one interchanges left and right (modules, invertibility etc.) i.e. .
is a -sided ideal in . The rings for which are called semiprimitive rings. In other words, for each nonzero element in a semiprimitive ring, by the definition, there is a simple module left annihilated by . Given any ring , the quotient is semiprimitive.