In abstract algebra, Jacobson's conjecture is an open problem in ring theory concerning the intersection of powers of the Jacobson radical of a Noetherian ring. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementIn a right-and-left Noetherian ring, the intersection of all nonnegative powers of the Jacobson radical J is {0}. In other words, the only element of a Noetherian ring in all powers of J is 0. 1 Proposed Year Progress Toward ResolutionVerified only for particular types of Noetherian rings: commutative Noetherian rings (a consequence of the Krull intersection theorem), fully bounded Noetherian rings, Noetherian rings with Krull dimension 1, and Noetherian rings satisfying the second layer condition. 1 Classification
Resolution Status Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Jacobson's conjecture (Wikipedia)
Sources
1. Jacobson's Conjecture (Wikipedia)
Wikimedia FoundationLead section
proposed in 1956 by Nathan Jacobson
Statement section
The only element of a Noetherian ring in all powers of J is 0.
Partial results section
Commutative Noetherian rings all satisfy Jacobson's conjecture. This is a consequence of the Krull intersection theorem.
View the Source Jacobson's conjecture (Wikipedia)
In Branch: Ring Theory, Lead sentenceQuote, In Branch: Ring Theory, Lead sentence
ebra, Jacobson's conjecture is an open problem in ring theory concerning the intersection of powers of the Jacobson radical of a N
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.