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Über Halbgruppen vertauschbarer Polynome. (On monoids of permutable polynomials). (German) Zbl 0664.13003
Let R be a commutative ring and X an indeterminate. This paper is concerned with the problem of determining the commutative submonoids of the set R[X] of polynomials considered as a monoid under composition \(f(X)\circ g(X)=f(g(X))\). Two subsets \(S_ 1\) and \(S_ 2\) of R[X] are said to be conjugate if there exists a linear polynomial \(h(X)=aX+b\) in R[X] such that a is a unit of R and \(S_ 2=\{h^{-1}(X)\circ g(X)\circ h(X)| g(X)\in S_ 1\}\) where \(h^{-1}(X)=a^{-1}(X-b)\). Let \(T=\{X+b| b\in R\}\), and if R is an integral domain with quotient field K let \(S_{\beta}=\{a(X-\beta)+\beta | a\in K\}\) for each \(\beta\in K.\)
Among the results given are the following: If R is an infinite integral domain and S is a maximal commutative subsemigroup of R[X] consisting of linear polynomials, then \(S=T\) or \(S=S_{\beta}\) for some \(\beta\in K\). The submonoids \(S_{\beta}\) are then each conjugate to \(S_ 0\) (and thus to each other) and T is not conjugate to \(S_ 0\). For another result let \(P_ n=\{aX^ k| k\equiv 1 (mod n)\), and a is an n-th root of unity in \(R\}\) for each natural number n. Then if R is an algebraically closed field the submonoid \(P_ n\) is maximal if and only if \(char(R)=0\) or \((n,char(R))=1\). Further, if S is a maximal commutative subsemigroup of R[X] which contains \(aX^ k\) for some \(a\in R-\{0\}\) and some \(k\geq 2\) which is not a power of the characteristic of R, then S is conjugate to \(P_ n\) for some n.
Reviewer: D.E.Rush
13B25 Polynomials over commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials