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Axiomatic definition of small cancellation rings. (English. Russian original) Zbl 1495.16023

Dokl. Math. 104, No. 2, 234-239 (2021); translation from Dokl. Ross. Akad. Nauk, Mat. Inform. Protsessy Upr. 500, 16-22 (2021).
The small cancellation (or generalizing small cancellation) theory for groups, i.e., when the common part between any two defining relations are small respect to each of them (for generalized small cancellation theory it is small respect to a biggest one), has provided machinery to solve numerous open problems in group theory (see [S. I. Adjan, The Burnside problem and identities in groups. Translated from the Russian by John Lennox and James Wiegold. Berlin-Heidelberg-New York: Springer-Verlag (1979; Zbl 0417.20001); S. I. Adyan, Russ. Math. Surv. 65, No. 5, 805–855 (2010; Zbl 1230.20001); translation from Usp. Mat. Nauk 65, No. 5, 5–60 (2010); A. Yu. Ol’shanskij, Geometry of defining relations in groups. Dordrecht etc.: Kluwer Academic Publishers (1991; Zbl 0732.20019)]). The control on ring relations is based on Gröbner basis and the breakthrough was provided by A. Smoktunowicz [Commun. Algebra 30, No. 1, 27–59 (2002; Zbl 1005.16021)].
The present paper deals with transferring the small cancellation concept for rings. Namely, in the present paper, the authors develop a small cancellation theory for associative algebras with a basis of invertible elements. They study quotients of a group algebra of a free group and introduce three specific axioms for the corresponding defining relations that provide the small cancellation properties for the obtained ring. First, the compatibility axiom is pretty natural, the notion of small piece is similar to the notion of small common relations fragment in group theory, (see condition 2 or the small cancellation axiom with constant \(\tau\ge 10\)). The specific theoretic conditions reflecting new ring theoretic properties are considered in the left isolation axiom or symmetrical right isolation axiom. It provides monomial control under natural transformations via relations. The authors show that this ring is nontrivial and it is called a small cancellation ring. If \(v=x^myx^{m+1}y\cdots x^n\) and \(w\) does not begin or end by \(x^{-1}, y^{-1}\) then the ring defined by the relation \(1+v+vw\) is a small cancellation ring.
Let us discuss the proof techniques (see [A. Atkarskaya et al., “Group-like small cancellation theory for rings”, Preprint, arXiv:2010.02836]) and its relations with classical ideas. Consider an ideal \(I\) with set of generators \(\{f_i\}\). The monomials are ordered firstly by length, then lexicographically. Let \(\bar{f_i}\) be the highest monomial of \(f_i\). The family \(\{f_i\}\) is called a Gröbner basis if in order to check whether \(g\in I\) a natural greedy algorithm is sufficient. It means that we take the highest term \(\bar{g}\) of \(g\), if it contains a submonomial of the form \(\bar{f_i}\) for some \(i\), we proceed with reduction, otherwise \(g\notin I\). Generally speaking, not every system of generators of an ideal is a Gröbner basis. In general, the problem of belonging to an ideal \(I\) is algorithmically unsolvable. It may happen that we have temporary increase the highest term in order to get a lower term in the final stage. A Gröbner basis of a finitely presented ideal can be infinite pretty often. However, there is a criterion for determining whether a set of generators of an ideal is a Gröbner basis or not. This criterion is provided by the Bergmans-Diamond lemma.
Let \(M=h\bar{f_i}=\bar{f_j}g\) such that the submonomials \(\bar{f_i}\) and \(\bar{f_j}\) of \(M\) overlap. Then \(M\) can be reduced by two ways (using reduction of the submonomial \(\bar{f_i}\) or submonomial \(\bar{f_j}\)) and take the difference \(f_{ij}\in I\) of the results. One can make further reductions of \(f_{ij}\) using a greedy algorithm for already the constructed \(f_i\) as long it is possible. The Diamond lemma states that if in any possible such proceeding we get zero, then \(\{f_i\}\) is a Gröbner-Shirshov basis. This can be looked as follows: if we have to rise in order to come lower, then in some place we come to a peak. The reduction condition of the Diamond lemma allows to cut it and the transform reduction algorithm is closer to the greedy one. There is a homological point of view on the Diamond lemma: if a transformation of one relation in our list by another one preserves the list. Then a natural greedy algorithm is sufficient. (see [L. Bokut et al., Gröbner-Shirshov bases. Normal forms, combinatorial and decision problems in algebra. Hackensack, NJ: World Scientific (2020; Zbl 1448.13001); G. M. Bergman, Adv. Math. 29, 178–218 (1977; Zbl 0326.16019)].)
A.I. Shirshov came with the concept of the Gröbner basis (it is more correct to call it Gröbner-Shirshov basis) in the process of proving the solvability of the equality problem in Lie algebras with one relation. A word is called regular if it is lexicographically larger than any of its cyclic conjugations. It turns out that in the correct word \( W \) one can unambiguously place the Lie brackets \([,]\) so that if they are expanded according to the rule \([a, b] = ab-ba\) then \(W\) will be the highest monomial of the resulting expression. Thus, a basis in a free Lie algebra is connected with regular words and is called a Hall-Shirshov basis. At the same time, the correct word cannot cling to itself. Therefore, the Gröbner basis in a universal enveloping algebra corresponding to a relation in a Lie algebra consists of one element. (Note that between cyclic conjugacy and the concept of regular words in Lie algebras and cyclic notation of relations in groups there is an obvious analogy, apparently not understood, see [A. I. Shirshov et al., Selected works of A. I. Shirshov. Translated by Murray Bremner and Mikhail V. Kotchetov. Edited by Leonid A. Bokut, Victor Latyshev, Ivan Shestakov and Efim Zelmanov. Basel: Birkhäuser (2009; Zbl 1188.01028)]).
The techniques of the paper have some analogies but it is much more advanced than the Gröbners-Shirshov basis technique. Instead of working with the deg-lex order (which was used by many authors according to my knowledge) the authors introduce some natural transformations. This notion is much more delicate and nontrivial than just usual lexicographical order descent. In order to proceed, the authors introduce charts and multyturns and present a detailed study of the influence of multyturns on the corresponding charts. They provide several invariants of charts, the simplest one is the number of its members.
Using these invariants, the authors introduce local filtrations and the corresponding graded objects. This grading replaces the notion of classical deg-lex order in the considerations. These graded objects allow an explicit description (see theorem 2). Using this description, they show that the quotient ring is not trivial, they find a linear basis and show that the equality problem is solvable by a greedy algorithm (an analog of Denns algorithm). They built a list of relations \(\mbox{Dp}\) and provide grading on these relations (like highest term of the element of classical Gröbner basis). They use our notions of derived monomials, function \(L\) (representing coming from monomial to smaller derived one), etc.
Then authors have to prove the set \(\mbox{Dp}\) is preserved under these relations. In other words, that \(\mbox{Dp}\) preserves reductions and this is the statement of the main lemma (which can be considered as a generalization of the homological core in the Bergmans-Diamond lemma):
Main lemma. Let \(U\) be an arbitrary monomial. Then \[\mbox{Dp}(\langle U\rangle_d) \cap L(\langle U\rangle_d) = \mbox{Dp}(L(\langle U\rangle_d)).\]
In other words, from dependencies with monomials one can come to dependencies with smaller (in appropriated sense) derived monomials and this process is under control. This means that the ideal of relations has a sui-generis Gröbner basis (with some complicated order on the set of monomials).
This is the first step to the Kurosh finitely generalized skew field problem. However, it seems that even this first step technique can be useful for numerous ring theoretical problems.

MSC:

16S34 Group rings
20F06 Cancellation theory of groups; application of van Kampen diagrams
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
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