Combinatorial Problems for Skew Fields I. Analogue of Britton's Lemma, and Results of Adjan‐Rabin Type

0. Introduction In [16] we proved the unsolvability of the word problem for skew fields, and proposed as a research programme that one should further pursue the analogy between groups and skew fields. The main result of the present paper is an analogue for skew fields of that central result of combinatorial group theory, Britton's lemma [3, 17]. Using this we sharpen [16] and obtain a precise analogue of the results of Adjan [2] and Rabin [18]. Our work is based on P. M. Cohn's beautiful studies of subrings of skew fields [4-12]. When we wrote [16] we were unaware of any notion of presentation for a skew field. Cohn [11] has since provided such a notion, which provides an interesting contrast to the notion of group presentation. We will make systematic use of this idea here. (A formulation of the present results in the style of [16] would be unintelligible algebraically.) To emphasize another debt to Cohn, we remark that to prove Britton's lemma for groups one has to solve a certain word problem [3,17]. Cohn's paper [8] gives the essential idea for doing the analogous thing for skew fields. We thank Professor Cohn and Professor W. W. Wheeler for reading an earlier version. Both detected a serious error, which is now corrected using a suggestion of Cohn. In addition, we have incorporated numerous suggestions of Cohn.