Department of Mathematics
Server

The Jacobson radical of a ring \( R \) has a handful of equivalent characterizations, amongst which \( J(R) \) is the ideal consisting of all elements of \( R \) which annihilate all simple right \( R \)-modules, the intersection of all maximal right ideals, and the left-module versions of the preceding. The Jacobson radical turns out to be an important tool in the study of the structure of rings and modules.

The (right transfinite) powers of the Jacobson radical are defined by \( J^{0}=R \); for ordinals \( \alpha > 0 \), \( J^{\alpha+1}=J^{\alpha}J(R) \); and for limit ordinals \( \lambda \), \( J^{\lambda}=\bigcap_{\alpha<\lambda}\,J^{\alpha} \). It follows easily from the Krull Intersection Theorem that if \( R \) is commutative noetherian, \( J^{\omega}=0 \). Jacobson [1] conjectured that the same was true of any (right) noetherian ring.

A counterexample soon followed from Herstein [2], and then another family of counterexamples from Jategaonkar [3], which were (right) principal ideal domains with the right ideals being precisely the chain of powers of the Jacobson radical.

Jacobson's conjecture is now taken to be about two-sided noetherian rings. It is known to be true for some classes more general than the commutative rings: fully bounded noetherian rings, noetherian rings with Krull dimension 1, noetherian rings satisfying the second layer condition. But the conjecture remains open in general,

1. Jacobson, Nathan (1956), Structure of rings, American Mathematical Society, Colloquium Publications, vol. 37, (p. 200), MR 0081264
2. Herstein, I. N. (1965), A counterexample in Noetherian rings, Proc. Nat. Acad. Sci. U.S.A. 54: 1036Ð1037
3. Jategaonkar, Arun Vinayak (1968), Left principal ideal domains, J. Algebra 8: 148Ð155,