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By Azriel Rosenfeld

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W. Boone at whose invitation this article was written. I. Adjan, On the work of Novikov and his pupils concerning algorithmic questions in algebra, Trudy Matem. Inst. A. Steklova AN SSSR 133 (1973), 23-32. A. Bokut’, Factorisation theorems for some classes of rings without zero divisors, 1. Algebra and Logic 4, No. 4 (1965), 25-52; 11, Ibid. 5 , No. 1 (1965), 5-30. A. Bokut’, On a property of Boone’s groups, Algebra and Logic 5 , No. 5 (1966), 5-23; 11, Ibid. 6, No. 1 (1967), 15-24. A. Bokut’, On Novikov’s groups, Algebra and Logic, 6, No.

J. Collins, Queen Mary College, London. A. Bokut' It is easy to show that the ring kQ contains no zero divisors and is not embeddable in a skew field. ) Theorem 1. The semigroup ( k Q ) * is embeddable in a group. The proof of this theorem appears in Bokut' [S], [9]. The idea of the proof is as follows. Since Q is given byhomogeneous relations, the ring kQ can be embedded in the ring kQ of infinite series. Then, to prove Theorem 1, it suffices to prove that t h e semigroup (kQ)*is embeddable in a group.

Suppose 'uqmqn f 1; if 1:; is cancelled in reducing T21qmqn to normal form, then, by Lemma 2, the word T ends in I;'C(ySy-'s,ll) or in I,kC(y-'SyI',ll), according as E' = 1 or E' = - 1. In either case Tq. is not normal. If 1:; is not cancelled, then R ends in I;'C(ySy-'2,J or I,kC(y-'SyI'ikl)and Rq, is not normal. For the group G4we have 'ur = V(Ly), '8, = v(Ly) where V(l,,y ) is a reduced and hence normal word. The set C4consists of all reduced words W K Uot'lUl * * * tekUk,where k a 0 and U, E C,, which contain no subwords (35) y v , lC(y-'SyV(y))t', I;'C(ySy-'V(y))t' where S = k 1 and S, as usual, is a reduced {sb}-word.

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An introduction to algebraic structures by Azriel Rosenfeld


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