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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
##### MSC:
 13B25 Polynomials over commutative rings 13F20 Polynomial rings and ideals; rings of integer-valued polynomials
##### Keywords:
composition of polynomials; linear polynomials