2012/02/29 by Pascal Schweitzer, Schweitzer, Pascal
Computer Science · Mathematics · #16S34 #20C07 (Primary) 68R05 (Secondary) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #cs.DM #math.GR #math.RA #msc:16S34 #msc:20C07 #msc:68R05 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1202.6645
20 pages, 8 figures
arxiv created 2012/02/29 · openalex publication_date 2012/02/29 · arxiv updated 2012/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Kaplanski's Zero Divisor Conjecture envisions that for a torsion-free group G and an integral domain R, the group ring R[G] does not contain non-trivial zero divisors. We define the length of an element a in R[G] as the minimal non-negative integer k for which there are ring elements r1,...,rk in R and group elements g1,...,gk in G such that a = r1 g1+...+rk gk. We investigate the conjecture when R is the field of rational numbers. By a reduction to the finite field with two elements, we show that if ab = 0 for non-trivial elements in the group ring of a torsion-free group over the rationals, then the lengths of a and b cannot be among certain combinations. More precisely, we show for various pairs of integers (i,j) that if one of the lengths is at most i then the other length must exceed j. Using combinatorial arguments we show this for the pairs (3,6) and (4,4). With a computer-assisted approach we strengthen this to show the statement holds for the pairs (3,16) and (4,7). As part of our method, we describe a combinatorial structure, which we call matched rectangles, and show that for these a canonical labeling can be computed in quadratic time. Each matched rectangle gives rise to a presentation of a group. These associated groups are universal in the sense that there is no counterexample to the conjecture among them if and only if the conjecture is true over the rationals.