In the next few years several other radicals were discovered, of which the most important example is the Jacobson radical.

The latter has various applications in the theory of Jacobson radicals.

It is a generalization of the Jacobson radical for rings.

The Jacobson radical is very similar to the nilradical in an intuitive sense.

The only idempotent contained in the Jacobson radical of a ring is 0.

However, a right quasiregular element need not necessarily be a member of the Jacobson radical.

The Jacobson radical of a ring has various internal and external characterizations.

Elements of the Jacobson radical of a ring, are often deemed to be "bad."

There are several equivalent characterizations of the Jacobson radical, such as:

Jacobson radical, the intersection of all maximal left ideals.

This is a ring whose Jacobson radical is zero.

In every quotient ring, the nilradical is equal to the Jacobson radical.

So, for commutative rings, the nilradical is contained in the Jacobson radical.

Therefore, the Jacobson radical also captures data which may seem to be not well-defined for noncommutative rings.

Elements of the Jacobson radical and nilradical can be therefore seen as generalizations of 0.

It is contained in, but in general not equal to, the ring's Jacobson Radical.

Every integral quotient ring of R has vanishing Jacobson radical.

This characterization of the Jacobson radical is useful both computationally and in aiding intuition.

Note, however, that in general the Jacobson radical need not consist of only the nilpotent elements of the ring.

If R is commutative, the Jacobson radical always contains the nilradical.

Rings with zero Jacobson radical are called semiprimitive rings.

A ring is called a semiprimitive ring if its Jacobson radical is zero.

If an element, r, of a ring is idempotent, it cannot be a member of the ring's Jacobson radical.

Many concepts from ring theory, such as ideals, Jacobson radicals, and factor rings can be generalized in a straightforward manner to this setting.

The radical of a module extends the definition of the Jacobson radical to include modules.