On the decidability of the zero-divisor problem

Details On the decidability of the zero-divisor problem On the decidability of the zero-divisor problem

2012-02-06

arxiv.org/abs/1202.1162v1

Mathematics

Group Theory

14 pages

Scientific paper

Let G be a finitely generated group geenerated by g_1,..., g_n. Consider the alphabet A(G) consisting of the symbols g_1,..., g_n 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 assumptions, 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 = (Z_2 \wr Z)^4 the zero-divisor problem is undecidable, in spite of the word problem being decidable.

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