abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

, Jacobson's conjecture is an open problem in

ring theory In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...

concerning the intersection of powers of the

Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definitio ...

of a

Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...

. It has only been proven for special types of Noetherian rings, so far. Examples exist to show that the conjecture can fail when the ring is not Noetherian on a side, so it is absolutely necessary for the ring to be two-sided Noetherian. The conjecture is named for the algebraist

Nathan Jacobson Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician. Biography Born Nachman Arbiser in Warsaw, Jacobson emigrated to America with his family in 1918. He graduated from the University of Alabama in 1930 and was awar ...

who posed the first version of the conjecture.

Statement

For a ring ''R'' with Jacobson radical ''J'', the nonnegative powers

J^n

are defined by using the product of ideals. :''Jacobson's conjecture:'' In a right-and-left

\bigcap_J^n=\.

In other words: "The only element of a Noetherian ring in all powers of ''J'' is 0." The original conjecture posed by Jacobson in 1956. As cited by . asked about noncommutative one-sided Noetherian rings, however Israel Nathan Herstein produced a counterexample in 1965, and soon afterwards, Arun Vinayak Jategaonkar produced a different example which was a left

principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are princip ...

. From that point on, the conjecture was reformulated to require two-sided Noetherian rings.

Partial results

Jacobson's conjecture has been verified for particular types of Noetherian rings: *

Commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

Noetherian rings all satisfy Jacobson's conjecture. This is a consequence of the

Krull intersection theorem In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...

. *

Fully bounded Noetherian ring Fully () is a municipality in the district of Martigny in the canton of Valais in Switzerland. History Fully is first mentioned in the 11th Century as ''Fuliacum''. Geography Fully has an area, , of . Of this area, 30.5% is used for agricultur ...

s * Noetherian rings with

Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generall ...

1 * Noetherian rings satisfying the second layer condition

References

Sources * * * * * * * * {{refend Conjectures Ring theory Unsolved problems in mathematics