Department of Mathematics
Server

Kegel (1963) proved that any ring which is a sum of two nilpotent subrings is itself nilpotent. He asked (1964) whether the sum of two nil subrings is also nil. This problem is related with the Koethe conjecture (1930) which states that a one-sided nil ideal of a ring is always contained in a two-sided nil ideal: Ferrero and Puczylowski (1989) established that the Koethe conjecture is equivalent to the statement that the sum of a nilpotent subring and a nil subring must be nil.

The first counterexample to the problem of Kegel was constructed by Kelarev (1993) and now there are several other counterexamples due to Salwa, Kelarev and Fukshansky in 1996-1999.

The purpose of the talk is to present an elementary exposition of the constructions of Kelarev, Salwa and Fukshansky and to give new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. All examples are monomial algebras. They can be considered as contracted semigroup algebras and the underlying semigroups are unions of locally nilpotent subsemigroups. In our constructions we make more transparent than in the past the close relationship between the considered problem and combinatorics of words. In particular, we show that any two-generated non-nilpotent monomial algebra of minimal growth can serve as a counterexample.

The talk is based on the joint work with Lakhdar Hammoudi "Combinatorics of words and semigroup algebras which are sums of locally nilpotent subalgebras" download paper