mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Köthe conjecture is a problem in ring theory,

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'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979 * ''Open'' (Go ...

. It is formulated in various ways. Suppose that ''R'' is a

ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

. 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 or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

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 all of its elements is nilpotent, i.e for each a \in I exists natural number ''n'' for which a^n = 0. If all elements of a ring ...

, other than , then it has no nil one-sided ideal, other than . 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 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 th ...

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 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 (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

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 a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

s with indeterminate ''x'' and

coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...

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 formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...

''R'' 'x'' # For any ring ''R'', the Jacobson radical of ''R'' 'x''consists of the polynomials with coefficients from the upper nilradical of ''R''. Necessary and sufficient conditions under which the statement 3 given above does not hold have been given recently.

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 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 "student John Smith is not lazy" is a c ...

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. Kegel proved that a ring which is the direct sum of two nilpotent

subring In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...

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

Survey paper (PDF)
{{DEFAULTSORT:Kothe Conjecture Ring theory Conjectures Unsolved problems in mathematics