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A unified approach to the Armendariz property of polynomial rings and power series rings. (English) Zbl 1165.16014
Let $$R$$ be a ring with 1, $$I$$ an ideal of $$R$$, $$[R;I][x]=R[x]+I[\![x]\!]$$ the subring of the power series ring $$R[\![x]\!]$$ of $$x$$ over $$R$$ where $$[R;I][x]=\{\sum_{i\geq 0}r_i x^i\in R[\![x]\!]\mid$$ there is some $$n$$ such that $$r_i\in I$$ for all $$i\geq n\}$$. Then $$R$$ is called $$I$$-Armendariz if whenever $$(\sum_{i\geq 0}a_ix^i)(\sum_{j\geq 0}b_jx^j)=0$$ in $$[R;I][x]$$, then $$a_ib_j=0$$ for all $$i$$ and $$j$$. Hence $$0$$-Armendariz is Armendariz and $$R$$-Armendariz is Armendariz of power series type. The authors extend some known results on Armendariz rings to $$I$$-Armendariz rings and obtain new results.
Theorem 1. A ring $$R$$ is $$I$$-Armendariz if and only if $$[R;I][x]$$ is $$I[\![x]\!]$$-Armendariz.
Also some Armendariz properties of several classes of $$I$$-Armendariz rings are given such as commutative rings and skew polynomial rings $$R[x;\sigma]/(x^{n+1})$$ for some integer $$n$$ where $$\sigma$$ is an injective endomorphism of $$R$$ with $$\sigma(1)=1$$. For an endomorphism $$\sigma$$ of $$R$$ with $$\sigma(I)\subset I$$, a ring $$R$$ is called $$(\sigma,I)$$-Armendariz if whenever $$(\sum_{i\geq 0}a_ix^i)(\sum_{j\geq 0}b_jx^j)=0$$ in $$[R;I][x;\sigma]$$, then $$a_i\sigma^i(b_j)=0$$ for all $$i$$ and $$j$$.
Theorem 2. Suppose that $$\sigma^{n_0}=\sigma$$ for some $$n_0>1$$. Then $$R$$ is $$(\sigma,I)$$-Armendariz if and only if $$[R;I][x;\sigma]$$ is $$(\sigma,I[\![x;\sigma]\!])$$-Armendariz.
Moreover, let $$M$$ be a right $$R$$-module, $$M[x]$$ (and $$M[\![x]\!]$$) all formal polynomials (power series) with coefficients in $$M$$, and $$N$$ a submodule of $$M$$, and $$[M;N][x]=M[x]+N[\![x]\!]$$. Then an $$R$$-module $$M$$ is called $$I$$-Armendariz if whenever $$m(x)f(x)=0$$ where $$m(x)=\sum m_ix^i\in[M;IM+MI][x]$$ and $$f(x)=(\sum a_jx^j)\in[R;I][x]$$, then $$m_ia_j=0$$ (resp. $$a_jm_i=0$$) for all $$i$$ and $$j$$. The ring $$R\propto M$$ ($$=\left(\begin{smallmatrix} a&m\\ 0&a\end{smallmatrix}\right)$$ for $$a\in R$$, $$m\in M$$) to be $$(I\propto N)$$-Armendariz is characterized where $$N=IM+MI$$.

##### MSC:
 16S36 Ordinary and skew polynomial rings and semigroup rings 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
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