In mathematics, the Köthe conjecture is a 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 ...

open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (Y ...

. It is formulated in various ways. Suppose that ''R'' is a ring. One way to state the

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...

is that if ''R'' has no

nil ideal In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent., p. 194 The nilradical of a commutative ring is an example of a nil ideal; in fact, it i ...

, other than , then it has no nil

one-sided ideal In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even number ...

, other than . This question was posed in 1930 by

Gottfried Köthe Gottfried Maria Hugo Köthe (born 25 December 1905 in Graz – died 30 April 1989 in Frankfurt) was an Austrian mathematician working in abstract algebra and functional analysis. Scientific career In 1923 Köthe enrolled in the University of ...

(1905–1989). The Köthe conjecture has been shown to be true for various classes of rings, such as

polynomial identity ring In ring theory, a branch of mathematics, a ring ''R'' is a polynomial identity ring if there is, for some ''N'' > 0, an element ''P'' ≠ 0 of the free algebra, Z, over the ring of integers in ''N'' variables ''X''1, ''X''2, ..., ''X'N'' such ...

s and right

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 ...

s, but a general solution remains elusive.

Equivalent formulations

The conjecture has several different formulations: # (Köthe conjecture) In any ring, the

sum Sum most commonly means the total of two or more numbers added together; see addition. Sum can also refer to: Mathematics * Sum (category theory), the generic concept of summation in mathematics * Sum, the result of summation, the additio ...

of two nil left ideals is nil. # In any ring, the sum of two one-sided nil ideals is nil. # In any ring, every nil left or right ideal of the ring is contained in the upper nil radical of the ring. # For any ring ''R'' and for any nil ideal ''J'' of ''R'', the

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

ideal M_''n''(''J'') is a nil ideal of M_''n''(''R'') for every ''n''. # For any ring ''R'' and for any nil ideal ''J'' of ''R'', the matrix ideal M₂(''J'') is a nil ideal of M₂(''R''). # For any ring ''R'', the upper nilradical of M_''n''(''R'') is the set of matrices with entries from the upper nilradical of ''R'' for every positive integer ''n''. # For any ring ''R'' and for any nil ideal ''J'' of ''R'', the

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...

s with indeterminate ''x'' and

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...

s from ''J'' lie in 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 the

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...

''R'' 'x'' # For any ring ''R'', the Jacobson radical of ''R'' 'x''consists of the polynomials with coefficients from the upper nilradical of ''R''.

Related problems

A conjecture by Amitsur read: "If ''J'' is a nil ideal in ''R'', then ''J'' 'x''is a nil ideal of the polynomial ring ''R'' 'x''" This conjecture, if true, would have

proven Proven is a rural village in the Belgian province of West Flanders, and a "deelgemeente" of the municipality Poperinge. The village has about 1400 inhabitants. The church and parish of Proven are named after Saint Victor. The Saint Victor ...

the Köthe conjecture through the equivalent statements above, however a

counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is ...

was produced by

Agata Smoktunowicz Agata Smoktunowicz FRSE (born 12 October 1973) is a Polish mathematician who works as a professor at the University of Edinburgh. Her research is in abstract algebra.

. While not a disproof of the Köthe conjecture, this fueled suspicions that the Köthe conjecture may be false. In , it was proven that a ring which is the direct sum of two nilpotent

subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...

s is itself nilpotent. The question arose whether or not "nilpotent" could be replaced with "locally nilpotent" or "nil". Partial progress was made when Kelarev produced an example of a ring which isn't nil, but is the direct sum of two locally nilpotent rings. This demonstrates that Kegel's question with "locally nilpotent" replacing "nilpotent" is answered in the negative. The sum of a nilpotent subring and a nil subring is always nil.Ferrero, M., Puczylowski, E. R., On rings which are sums of two subrings, Arch. Math. 53 (1989), p4–10.

References

External links

PlanetMath page

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{{DEFAULTSORT:Kothe Conjecture Ring theory Conjectures Unsolved problems in mathematics