The problem we consider is when a group ring K[G] over a field is reversible, i.e. satisfies the implication (ab = 0) → (ba = 0). For G torsion-free, this is strictly connected with the zero divisor conjecture. In this paper, we characterize reversible rings K[G] for torsion groups. In particular, all finite reversible group rings are described. Our results exhibit a broad class of reversible rings, which are not symmetric.

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a b s t r a c tThe following problem looking as a high-school exercise hides an unexpected difficulty. Do the matrices A = 2 0 0 3 and B = 3 5 0 5 satisfy any nontrivial equation with the multiplication symbol only? This problem was mentioned as open in Cassaigne et al. [J. Cassaigne, T. Harju, J. Karhumäki, On the undecidability of freeness of matrix semigroups, Internat. J. Algebra Comput. 9 (3-4) (1999) 295-305] and in a book by Blondel et al. [V. Blondel, J. Cassaigne, J. Karhumäki, Problem 10.3: Freeness of multiplicative matrix semigroups, in: V. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton University Press , 2004, pp. 309-314] as an intriguing instance of a natural computational problem of deciding whether a given finitely generated semigroup of 2 × 2 matrices is free. In this paper we present a new partial algorithm for the latter which, in particular, easily finds that the following equation holds for the matrices above. 1 Our algorithm turns out quite practical and allows us to settle also other related open questions posed in the mentioned article.