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• W2026857107 abstract "Let R be the free associative algebra on some set of generators over a field k (equivalently: the tensor algebra on some k-vector-space). Though R is in general very noncommutative, it is easy to find pairs of commuting elements: If we take any z E1 R, and polynomials P and Q in one indeterminate over k, then P(z) and Q(z) will commute. We shall here show that the centralizer, C, of any nonscalar element u E1 R (i.e. C={x E R 1 xu=ux}) is of the form k[z] for some nonscalar z E In particular, any pair of commuting elements of R be written in the form P(z), Q(z). (This was conjectured by P. M. Cohn [8, p. 348].) Let us first outline our proof, stating results as they apply to this problem, and not necessarily in the greatest generality in which they will be proved: After setting up some general ring-theoretic tools in ?1, we shall show in ?2, by rather elementary arguments that our centralizer ring C is commutative, and in fact is a finite integral extension of k[u]. To complete our proof, it suffices to show that C is integrally closed, and is embeddable in a polynomial ring k[x], because any subalgebra of a polynomial algebra k[x] that is integrally closed (in its own field of fractions) and /k is of the form k[y] [9, Proposition 2.1]. (The proof uses Liuroth's theorem.) In ?3 we find that there exists a nonzero element e E1 R, and a homomorphismf of the integral closure C' of C, into R, such that for all x E C, xe=ef(x). In other words, f(C) is conjugate to C, and this conjugate, if not C itself, can be integrally closed within R. In ?4, we show thatf(C') be 'pulled back across e, thus C itself has integral closure in This integral closure, being commutative and containing u, must coincide with our centralizer C. So C is integrally closed. In ?5, we show that any finitely generated nontrivial (i.e., #Ak) subalgebra of R be mapped into a polynomial ring k[x] so as to have nontrivial image. We apply this to C. Now the image will have transcendence degree I over k, and C has the same transcendence degree because it is a finite extension of k[u]. Hence the map will be 1-1 (see [16, Chapter II, ?12, Theorem 29, p. 101]), giving the embedding required to complete our proof. The remaining ??6-9, examine some related points-centralizers in some rings of differential operators (?9), generalizations of results of the preceding sections, and" @default.
• W2026857107 created "2016-06-24" @default.
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• W2026857107 date "1969-01-01" @default.
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• W2026857107 title "Centralizers in free associative algebras" @default.
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• W2026857107 doi "https://doi.org/10.1090/s0002-9947-1969-0236208-5" @default.
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