\(\mathfrak{m}\)-full and basically full ideals in rings of characteristic \(p\)
Let \((R,m)\) be a local noetherian ring. An ideal \(I\) of \(R\) is said to be \(m\)-full if there exists \(x \in m \setminus m^2\) such that \((Im:x)=I\). Originally introduced by \textit{J. Watanabe} [Nagoya Math. J. 106, 101--111 (1987; Zbl 0623.13012)], the \(m\)-full ideals have been used as a tool in the study of the structure of integrally closed ideals. Heinzer, Ratliff and Rush [\textit{W. J. Heinzer} et al., J. Algebra 250, No. 1, 371--396 (2002; Zbl 1048.13001)] introduced the following related concept: an ideal \(I\) is said to be basically full if for every ideal \(J\) containing \(I\) no minimal set of generators of \(I\) can be extended to a minimal set of generators for \(J\). (Every \(m\)-full ideal is basically full.) In this paper the author introduces two tight closure versions of \(m\)-full and basically full ideals. Assuming that \(R\) has prime characteristic \(p\), an ideal \(I\) is said to be \(*\)-\(m\)-full if \((Im:x)=I^{*}\) for some \(x \in m \setminus m^2\), where \(I^{*}\) denotes the tight closure of \(I\). An ideal \(I\) is said to be weakly \(*\)-\(m\)-full if \((Im:x)^{*}=I^{*}\) for some \(x \in m \setminus m^2\). Similarly, \(I\) is said to be \(*\)-basically full if \((Im:m)=I^{*}\) and \(I\) is said to be weakly \(*\)-basically full if \((Im:m)^{*}=I^{*}\). Several properties of these concepts are studied as well as their connections with the original versions. In particular, it is shown that every \(m\)-full (basically full) ideal is weakly \(*\)-\(m\)-full (\(*\) basically full). However, there exist \(*\)-\(m\)-full ideals which are not \(m\)-full.
scientific article; zbMATH DE number 6339722
scientific article