Strongly bounded rings
The main result of this paper is the following condition for rings in which tight closure commutes with localization. Let \(R\) be a commutative noetherian ring of characteristic \(p\). Then \(R\) should have a test element \(c\) such that there exists an \(R\)-linear map \(\theta: R^{1/q} \to R^{1/pq}\) taking \(c^{1/q}\) to \(c^{1/pq}\) for some power \(q= p^e\). This condition also gives an effective bound for computing tight closure. The author terms such a ring \(R\) strongly bounded with respect to \(c\). She goes on to show that any \(F\)-finite domain \(R\) whose integral closure \(S\) is regular is strongly bounded with respect to an element \(c \in R \setminus \{0\}\) such that \(cS \subseteq R\). In particular, this includes the case of one-dimensional \(F\)-finite domains.
scientific article; zbMATH DE number 1117788