Köthe conjecture
Encyclopedia
In mathematics
, the Köthe conjecture is a problem in ring theory
, open . It is formulated in various ways. Suppose that R is a ring
. One way to state the conjecture is that if R has no nil ideal
, other than {0}, then it has no nil one-sided ideal, other than {0}. This question was posed in 1930 by Gottfried Köthe
(1905–1989). The Köthe conjecture has been shown to be true for various classes of rings, such as polynomial identity rings, but a general solution remains elusive.
An equivalent is that the sum of two left nil ideals is a nil ideal. Kegel (1964) asked whether the sum of two nil subrings is also nil. A counterexample to this was found by Kelarev (1993). It is known that the original conjecture is equivalent to the statement that sum of a nilpotent subring and a nil subring is always nil.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Köthe conjecture is a problem in ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, open . It is formulated in various ways. Suppose that R is a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
. One way to state the conjecture is that if R has no nil ideal
Nil ideal
In mathematics, more specifically ring theory, an ideal of a ring is said to be a nil ideal if each of its elements is nilpotent. The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil...
, other than {0}, then it has no nil one-sided ideal, other than {0}. This question was posed in 1930 by Gottfried Köthe
Gottfried Köthe
Gottfried Maria Hugo Köthe was an Austrian mathematician working in abstract algebra and functional analysis.- Scientific career :...
(1905–1989). The Köthe conjecture has been shown to be true for various classes of rings, such as polynomial identity rings, but a general solution remains elusive.
An equivalent is that the sum of two left nil ideals is a nil ideal. Kegel (1964) asked whether the sum of two nil subrings is also nil. A counterexample to this was found by Kelarev (1993). It is known that the original conjecture is equivalent to the statement that sum of a nilpotent subring and a nil subring is always nil.