On the decidability of the zero-divisor problem
Łukasz Grabowski · arXiv (Cornell University) · 2012
Abstract. Let G = 〈g1,..., gn 〉 be a finitely generated group. Consider the alphabet A(G) consisting of the symbols g1,..., gn and the symbols “+ ” and “−”. The words in this alphabet represent elements of the integral group ring Z[G]. We investigate the computational problem of deciding whether a word in the alphabet A(G) determines a zero-divisor in Z[G]. Under suitable assump-tions, we observe that the decidability of the word problem for G implies the decidability of the zero-divisor problem. However, we show that in the group G = (Z2 oZ)4 the zero-divisor problem is undecidable, in spite of the word problem being decidable. 1.