Graded almost valuation domains (Q2244698)

Let \(\Gamma \) be a commutative and cancellative monoid (written additively) whose quotient group \(\langle \Gamma \rangle \) = \(\{a-b|a,b\in \Gamma \}\), is a torsionfree abelian group. It is well known that a cancellative monoid \( \Gamma \) is torsionless if and only if \(\Gamma \) can be given a total order compatible with the monoid operation. A \(\Gamma \)-graded integral domain \(R\) , is a direct sum \(R=\bigoplus _{\alpha \in \Gamma }R_{\alpha }\) of subgroups \( R_{\alpha }\) of \(R\) such that \(R_{\alpha }R_{\alpha }\subseteq R_{(\alpha +\alpha])}\) for every \(\alpha\ ,\alpha] \in \Gamma .\) An element \(x\) of a graded domain is called homogeneous of grade \(\alpha \) if \(x\in R_{\alpha }\) . So the \(0\) of \(R\) is of every grade and the \(1\) of \(R\) is of grade \(0.\) (Here \(0\) is the identity of \(\Gamma.)\) It is easy to see that \(R_{0}\) is a subring of \(R\) and each of \(R_{\alpha }\) is an \(R_{0}\)-module. The set \( G_{0}=\Gamma \cap -\Gamma \) is the group of units of \(\Gamma .\) Let \(H\) be the set of nonzero homogeneous elements of \(R.\) Obviously \(H\) is a saturated multiplicative set in \(R\) and we can form a ring of fractions \(R_{H}.\) Now \( R_{H}=\) \(\oplus _{\gamma \in \langle \Gamma \rangle }(R_{H})_{\gamma }\) with each \((R_{H})_{\gamma }=\{a/b|a\in R_{\alpha },0\neq b\in R_{\beta }\) and \( \gamma =\alpha -\beta \}.\) In particular, \((R_{H})_{0}\) is a field, and each nonzero homogeneous element of \(R_{H}\) is a unit. An overring \(T\), with \( R\subseteq T\subseteq R_{H}\) is called a homogeneous overring if \(T= \bigoplus _{\gamma \in \langle \Gamma \rangle }(T\cap (R_{H})_{\gamma }).\) A subring \(S\) of \(R\) is called a homogeneous subring if the homogeneous components of every element of \(S\) belong to \(S\). An ideal \(I\) of \(R\) is a homogeneous ideal if \(I\) can be generated by homogeneous elements. Let \( T=\oplus _{\alpha \in \Gamma }T_{\alpha }\) be a graded ring and \(R\) a homogeneous subring of \(T\) . The extension \(R\subseteq T\) is called a graded root extension (gr-root extension) if for every homogeneous element \(x\in T\) , there exists an integer \(n=n(x)\geq 1\) with \(x^{n}\in R\). It is common, these days, to take up a certain type, say \(X,\) of integral domains and study their graded version calling it a gr-\(X\) domain, e.g., gr-Prufer domain may be described as a graded domain in which every nonzero homogeneous ideal of \(R\) generated by finitely many homogeneous elements is invertible. Likewise a gr-valuation domain is a graded domain \(R\) such that for every pair \(x,y\) of homogeneous elements we have \(x|y\) or \(y|x.\) \textit{D. D. Anderson} and \textit{M. Zafrullah} [J. Algebra 142, No. 2, 285--309 (1991; Zbl 0749.13013)] called a domain \(R\) an almost valuation domain (AVD) if for each pair of nonzero elements \(x,y\) there is a positive integer \(n=n(x)\) such that \(x^{n}|y^{n}\) or \(y^{n}|x^{n},\) or equivalently for each nonzero ~\(x\) in \(qf(R)\) there is a positive integer \(n\) such that \(x^{n}\in R\) or \(x^{-n}\in R.\) Similarly, one can define an additive monoid \(\Lambda \) as an almost valuation monoid if for every \(x\in \langle \Gamma \rangle \) there is a positive integer \(n\) such that \(nx\in \Gamma \) or \(-n(x)\in \Gamma .\) The authors of the paper under review have taken up what may be termed as gr-AVD. They show that \(R\) is a gr-AVD if and only if the following conditions hold. 1. \(\Gamma \) is an almost valuation monoid, 2. \(K\subseteq (RH)_{0}\) is a root extension, 3. If \(\alpha \in \Gamma \) is not a unit, then for every \( 0\neq x\in R_{\alpha }\) and \(0\neq r\in R_{0}\), \(r^{n}|x^{n}\) in \(R\) for some \(n\geq 1\), and 4. \(T=\) \(\bigoplus _{\gamma \in G_{0}}(R_{H})_{\gamma }\) is a gr-AVD. The paper includes some interesting examples as well.