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• W2012303062 abstract "It is proved that the multiplicative semigroup of the ring of polynomials in two commuting indeterminates over a noncommutative domain contains a noncommutative free subsemigroup. The question whether the multiplicative group of a division ring D contains a free noncommutative subgroup has been raised by Lichtman [4]. The answer is positive when D is finite-dimensional over its center Z. A more general result is given in [2]. The question is studied further in [3], but the general case is still open. Makar-Limanov raised a seemingly simpler question: Does D* = D {O} contain a free noncommutative subsemigroup? In [5] he proved that the answer is positive when Z is uncountable. In the present note we propose a more general question: Given a noncommutative domain D (with 1) does D* contain a free noncommutative subsemigroup? Using a recent result [1, Theorem 3] it is possible to extend MakarLimanov's result to noncommutative domains. We prove a stronger result that has the advantage that it does not make any assumptions on the center. It says that if R is a noncommutative domain and u, v are commuting indeterminates over R, then the multiplicative semigroup of the ring of polynomials R[u, v] contains a free noncommutative subsemigroup. The case of a domain with uncountable center then becomes an easy corollary. Let W be the free semigroup of words in two letters x, y. A nonempty word can be written in the form xilyjl ... xiryIr with r, ji, ... , ir > 1 and ii, jr > 0. A relation between two elements a, b of a semigroup is a pair (wI, w2) of two distinct words wI, w2 E W. Since we are interested in the semigroup S* where S is a domain, we may consider only relations (wI, w2) such that w1, w2 do not end (and do not start) with the same letter. A relation is said to be homogeneous if w1 and w2 have the same degree in x and the same degree in y. In what follows, R will be a domain and R[u, v] the polynomial ring in two commuting indeterminates u, v over R. The subring of R generated by 1 is denoted by P, and P(x, y) is the free algebra in x, y over P. Received by the editors October 12, 1990 and, in revised form, March 25, 1991. 1991 Mathematics Subject Classification. Primary 16U10; Secondary 20M05." @default.
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• W2012303062 date "1992-02-01" @default.
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• W2012303062 title "Free subsemigroups of domains" @default.
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• W2012303062 doi "https://doi.org/10.1090/s0002-9939-1992-1096212-2" @default.
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