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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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