Fuzzy semistar operations on integral domains
The author introduces the notion of a fuzzy semistar operation on an integral domain and relates it to the existing notion of a fuzzy star operation. He shows that the set of all fuzzy semistar operations on the integral domain is a complete lattice. He also characterizes when an overring of a domain is a Prüfer domain and when an overring is a flat module over the domain in terms of a fuzzy semistar operation. Let \(R\) be an integral domain. Let \(F_z(R)\) denote the set of all fractionary fuzzy ideals of \(R\) and \(\overline F_z(R)\) the set of all fuzzy \(R\)-submodules of \(K\), where \(K\) is the quotient field of \(R\). A fuzzy semistar operation on \(R\) is a function \(*:\overline F_z(R)\to\overline F_z(R)\) such that the following properties hold for all \(\alpha,\beta\in\overline F_z(R)\) and \(0\neq d\in K\): \((*_1)\) \((d_1\circ\beta)^*=d_1\circ\beta^*\); \((*_2)\) \(\alpha\subseteq\beta\Rightarrow \alpha^*\subseteq\beta^*\); \((*_3)\) \(\beta\subseteq\beta^*\) and \(\beta^{**}=\beta^*\). In addition to the results stated above, the author examines the extensions of fuzzy star operations to fuzzy semistar operations and when a fuzzy semistar restricts to a fuzzy star operation. The author also fuzzifies some classical results of multiplicative ideal theory with the use of a fuzzy semistar operation. He shows that an overring \(T\) of \(R\) is a flat \(R\)-module if and only if the fuzzy star operation \(*_T\) is stable.
scientific article; zbMATH DE number 6112280