Showing papers on "Free algebra published in 1971"

Automorphisms of a free associative algebra of rank 2

Abstract: Let Ä<2> = R that keeps (xy — yx) fixed (up to multiplication by an element of R), then

. This follows from a more general result about endomorphisms of Ä<2> via a theorem due to H. Jung [6] concerning automorphisms of a commutative and associative algebra of rank 2. Introduction. Let F be a commutative domain with 1. We will further assume that every invertible matrix with coefficients in F is a product of elementary matrices (e.g. F a field, F a Euclidean domain). Let F be the free associative algebra of rank n over F. It has been conjectured (see P. M. Cohn [1, p. 33]) that every F automorphism of F is tame (i.e. a product of elementary automorphisms of F). The conjecture has been proved true for a free Lie algebra by P. M. Cohn [3], and for R(x,y), the free commutative and associative algebra of rank 2 (i.e. the polynomial ring in 2 commuting indeterminates x and y) by Jung [6]. We shall prove here that a certain class of automorphisms of F<2> satisfies the conjecture, namely that if F so that [A, B] = X[x, y], X a unit of F, is it true that A and B generate F<2>? We prove that the answer is \"yes\" if A=x + P, B=y+ Q, where F and Q satisfy condition (iii) below. The following is an outline of our proof. Given

Embedding free algebras in skew fields

Abstract: This paper constructs a minimal element in the partial order on the set of skew fields generated by a free algebra, and shows that the partial order contains a certain sub partial order. Examples of embedding free algebras in skew fields of heights one and two are also given. Let R be an integral domain (noncommutative) with identity. We will assume that the integral domain R is always embedded in some skew field (this need not be the case in general [2], [3], [6], [7]). There are many skew fields which contain R but in the commutative case there is only one field which is generated by R, namely the field of fractions. In the noncommutative case there may be several distinct skew fields generated by R [4, pp. 277]. For skew fields D1 and D2 generated by R, we say D1 > D2 if there exists a place from D1 to D2 which extends the natural isomorphism between the embeddings of R. This paper shows that this is a partial order on the set (P (we identify isomorphic embeddings) of skew fields generated by R and in the case where R is the free algebra on two generators we show that (P contains the subposet ? where C(x) is the unique maximal element [1, Theorem 27 ] of (P and Ki are distinct elements of (P with K2 minimal in (P. We also examine the height of an integral domain and give examples of embeddings of different heights of a free algebra in two skew fields. I. The partial order and the chain of domains Qi(K, ?(R)). DEFINITION. Let D be a skew field and 0 an isomorphism of R into D. R is fully embedded in D if the smallest sub skew field of D containing + (R) is D itself. We will denote a full embedding of (D, 4 (R)) by (D, q). DEFINITION. If D1 and D2 are two division rings we say 0 is a place from D1 to D2 if 0 is a homomorphism from a local subring S of D, onto D2 [local means the set of nonunits is an ideal ]. DEFINITION. For full embeddings (D, a) and (K, y) we say (D, a) > (K, Py) if there is a place q from D to K such that qV'(K) Doa(R) and f1 a (R) =,y-c'. Received by the editors October 9, 1970. AMS 1969 subject classifications. Primary 1646, 1615.