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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 \(n\).

MSC:
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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