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A unified approach to various generalizations of Armendariz rings. (English) Zbl 1198.16025
Let \(R\) be a ring, \((S,\leq)\) a strictly ordered monoid (i.e., for all \(a,b,c\in S\), \(a<b\) implies \(ca<cb\) and \(ac<bc\)), and \(\omega\colon S\to\text{End}(R)\) a monoid homomorphism denoted by \(\omega_s\) for \(s\in S\). Let \(A\) be the set of all functions \(f\colon S\to R\) such that the support of \(f\), \(\{s\in S\mid f(s)\neq 0\}\) is Artinian and narrow (i.e., every nonempty subset of \(S\) has at least one but only a finite number of minimal elements). Denote the set \(\{(x,y)\in\text{supp}(f)\times\text{supp}(g)\mid s=xy\}\) by \(X_s(f,g)\) for \(s\in S\) and \(f,g\in A\), and define \((fg)(s)=\sum f(x)\omega_s(g(y))\) for \((x,y)\in X_s(f,g)\) and \((f+g)(s)=f(s)+g(s)\). Then \(A\) is called the ring of skew generalized power series \(R[\![S,\omega,\leq]\!]\) with coefficients in \(R\) and exponents in \(S\).
The ring \(A\) is a generalization of a skew polynomial ring, skew power series ring, skew Laurent polynomial ring, and skew group ring. A ring is called Armendariz relative to \(S\) if \(f=\sum a_is_i\) and \(g=\sum b_it_i\in A\) such that \(fg=0\) implies \(a_ib_j=0\) for all \(i,j\), and \(R\) is called \((S,\omega)\)-Armendariz if \(fg=0\) for \(f,g\in A\) implies \(f(s)\omega_s(g(t))=0\) for all \(s,t\in S\). The class of \((S,\omega)\)-Armendariz rings generalizes skew Armendariz rings. An \(S\)-compatible (or \(S\)-rigid) ring \(R\) is defined if \(\omega_s\) is compatible (or rigid) for all \(s\in S\). Then the authors show some properties of an \(S\)-compatible ring in terms of the annihilators of subsets \(Y\subset A\) and \(X\subset R\) in \(R\).
Theorem. Assume \(R\) is compatible. If \(A\) is right Ikeda-Nakayama, so is \(R\) (i.e., \(\text{ann}^R_l(I\cap J)=\text{ann}^R_l(I)+\text{ann}^R_l(J)\) for all right ideals \(I,J\) of \(R\) where \(\text{ann}^R_l(X)=\) all left annihilators of \(X\) in \(R\)).
Also properties of an \((S,\omega)\)-Armendariz ring are given related to other kinds of rings such as reduced, reversible, semicommutative, Abelian, 2-primal, and semiprime Goldie. Moreover, an \((S,\omega)\)-Armendariz ring \(R\) is characterized in terms of the extensions \(R[\![x,\sigma]\!]/(x^n)\), \(R[\![x]\!]\), \(R[x,x^{-1}]\), and \(R[\![x,x^{-1}]\!]\) where \(\sigma\) is an injective endomorphism of \(R\) and \(n\geq 2\). Furthermore, it is shown that some kind of right uniserial rings are \((S,\omega)\)-Armendariz, and an \((S,\omega)\)-Armendariz ring \(R\) is characterized in terms of subring of an upper triangular matrix ring of order \(n\) over \(R\) for any integer \(n\).

16S36 Ordinary and skew polynomial rings and semigroup rings
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16U80 Generalizations of commutativity (associative rings and algebras)
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