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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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