\(t\)-splitting sets in integral domains.
This paper investigates various aspects of \(t\)-splitting sets in integral domains. Let \(D\) be an integral domain with quotient field \(K\). For a nonzero fractional ideal \(I\) of \(D\), \(I^{-1}=\{x\in K/ xI\subseteq D\}\), \(I_{v}=(I^{-1})^{-1}\), and \(I=\bigcup\{(a_{1},\dots,a_{n})_{v}/\;\;\; (0)\neq(a_{1},\dots,a_{n})\subseteq I\}\), also \(I\) is a divisorial ideal or \(v\)-ideal (resp., \(t\)-ideal) if \(I_{v}=I\) (resp., \(I_{t}=I\)). A multiplicative subset \(S\) of \(D\) is \(t\)-splitting set if for each \(0\neq d\in D\), \(dD=(AB)_{t}\) for some integral ideals \(A\) and \(B\) of \(D\), where \(A_{t}\cap sD=sA_{t}\) for all \(s\in S\) and \(B_{t}\cap S\neq\emptyset\). A \(t\)-splitting set \(S\) of \(D\) is a \(t\)-lcm (resp., Krull) \(t\)-splitting set if \(sD\cap dD\) is \(t\)-invertible (resp., \(sD\) is a \(t\)-product of height-one prime ideals of \(D\)) for all nonunits \(s\in S\) and \(0\neq d\in D\). Let \(S\) be a \(t\)-splitting set of \(D\), \(\mathcal{T}=\{A_{1}\dots A_{n}/ A_{i}=d_{i}D_{S}\cap D \;\;\;\text{ for\;some\;}0\neq d_{i}\in D\}\), and \(D_{\mathcal{T}}=\{x\in K/xC\subseteq D\;\;\text{ for\;some\;}C\in \mathcal{T}\}\). Recall that an integral domain \(D\) is a UMT-domain if every upper to zero in \(D[X]\) is a maximal \(t\)-ideal. The following result gives another characterization of a UMT-domain using the notion of \(t\)-splitting: Let \(D\) be an integral domain. Then \(D\diagdown\{0\}\) is a \(t\)-splitting set in \(D[X]\) if and only if \(D\) is a UMT-domain. The following result gives the link between UMT-domain and PVMD: Let \(D\) be an integral domain. If \(D\) is a UMT-domain, then \(D\) is a PVMD if and only if for every \(0\neq f\in D[X]\), \(fD[X]=(A_{f}B)_{t}\), where \(B\) is the \(t\)-product of uppers to zero. The second part of this paper treats a new concept ``\(t\)-lcm \(t\)-splitting set'' which is a generation of a \(t\)-lcm splitting set. In this sense, the authors show that: Let \(D\) be an integral domain with quotient field \(K\), \(S\) a \(t\)-splitting set of \(D\), \(\mathcal{T}=\{A_{1}\dots A_{n}/ A_{i}=d_{i}D_{S}\cap D \) for some \(0\neq d_{i}\in D\}\). Then the following statements are equivalent (i) \(S\) is a \(t\)-lcm \(t\)-splitting set; (ii) Every finite type integral \(v\)-ideal of \(D\) intersecting \(S\) is \(t\)-invertible. (iii) \(D_{\mathcal{T}}=\{x\in K/xC\subseteq D\) for some \(C\in \mathcal{T}\}\) is a PVMD. As a corollary, \(D\) is a UMT-domain if \(D_{S}\) is UMT-domain. This work is concluded by an interesting result: \(D\diagdown\{0\}\) is a \(t\)-splitting set in \(D[X]\) if and only if \(D\) is a UMT-domain and that every nonempty multiplicative subset of \(D[X]\) contained in \(G=\{f\in D[X]/(A_{f})_{v}=D\}\) is a \(t\)-lcm \(t\)-complemented \(t\)-splitting set of \(D[X]\).
scientific article; zbMATH DE number 2035894
scientific article