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Combinatorics of words and semigroup algebras which are sums of locally nilpotent subalgebras. (English) Zbl 1069.16020

In 1964, O. H. Kegel [J. Algebra 1, 103–109 (1964; Zbl 0203.04201)] asked whether a ring which is a sum of two locally nilpotent subrings is itself locally nilpotent. In 1993, A. V. Kelarev [Arch. Math. 60, No. 5, 431–435 (1993; Zbl 0784.16011)] gave the first counterexample to this problem: A ring which is a sum of two locally nilpotent subrings may not be nilpotent. Later A. Salwa [Commun. Algebra 24, No. 12, 3921–3931 (1996; Zbl 0878.16012); ibid. 25, No. 12, 3965–3972 (1997; Zbl 0903.16016)] and A. Fukshansky [Proc. Am. Math. Soc. 128, No. 2, 383–386 (2000; Zbl 0983.16014)] constructed other examples of this kind.
Here the authors generalize in a certain way all these examples using combinatorics of words. The last section is devoted to a generalization of A. T. Kolotov’s semigroup algebra [see Algebra Logika 20, 138–154 (1981; Zbl 0535.16020)].

MSC:

16N40 Nil and nilpotent radicals, sets, ideals, associative rings
68R15 Combinatorics on words
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
20M05 Free semigroups, generators and relations, word problems
20M25 Semigroup rings, multiplicative semigroups of rings
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