Elements of the Jacobson radical and nilradical can be therefore seen as generalizations of 0. Equivalent characterizations The Jacobson radical of a ring has various internal and external characterizations. Source: Internet
In fact for any ring, the nilpotent elements in the center of the ring are also in the Jacobson radical.sfn So, for commutative rings, the nilradical is contained in the Jacobson radical. Source: Internet
Isaacs, Corollary 13.4, p. 180 sfn Alternatively, one could replace "right" with "left" in the previous sentence.sfn This characterization of the Jacobson radical is useful both computationally and in aiding intuition. Source: Internet
More generally: the Jacobson radical of every local ring is the unique maximal ideal of the ring. Source: Internet
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. Source: Internet
The following are equivalent characterizations of the Jacobson radical in rings with unity (characterizations for rings without unity are given immediately afterward): * J(R) equals the intersection of all maximal right ideals of the ring. Source: Internet