Jacobson Radical - Intuitive Discussion

Intuitive Discussion

As with other radicals of rings, the Jacobson radical can be thought of as a collection of "bad" elements. In this case the "bad" property is that these elements annihilate all simple left and right modules of the ring. For purposes of comparison, consider the nilradical of a commutative ring, which consists of all elements which are nilpotent. In fact for any ring, the nilpotent elements in the center of the ring are also in the Jacobson radical. So, for commutative rings, the nilradical is contained in the Jacobson radical.

The Jacobson radical is very similar to the nilradical in an intuitive sense. A weaker notion of being bad, weaker than being a zero divisor, is being a non-unit (not invertible under multiplication). The Jacobson radical of a ring consists of elements which satisfy a stronger property than being merely a non-unit – in some sense, a member of the Jacobson radical must not "act as a unit" in any module "internal to the ring." More precisely, a member of the Jacobson radical must project under the canonical homomorphism to the zero of every "right division ring" (each non-zero element of which has a right inverse) internal to the ring in question. Concisely, it must belong to every maximal right ideal of the ring. These notions are of course imprecise, but at least explain why the nilradical of a commutative ring is contained in the ring's Jacobson radical.

In yet a simpler way, we may think of the Jacobson radical of a ring as method to "mod out bad elements" of the ring – that is, members of the Jacobson radical act as 0 in the quotient ring, R/J(R). If N is the nilradical of commutative ring R, then the quotient ring R/N has no nilpotent elements. Similarly for any ring R, the quotient ring has J(R/J(R))={0} and so all of the "bad" elements in the Jacobson radical have been removed by modding out J(R). Elements of the Jacobson radical and nilradical can be therefore seen as generalizations of 0.