Construction and finite generation of the strict closure of rings. (English) Zbl 07842166
Summary: The construction of Arf rings and strictly closed rings has been studied widely; however, there has been no clear description of the structure of the strict closure \(R^{\ast}\) when \(\overline{R}\) is not a finitely generated \(R\)-module. In this paper, we investigate the construction and finite generation of the strict closure of rings. We determine its structure when \(R\) is a Cohen-Macaulay semi-local ring of dimension one, with \(\dim R_M = 1\) for every \(M \in \operatorname{Max} R\). Using this, a characterization of the finite generation of the strict closure is given.
MSC:
| 13A15 | Ideals and multiplicative ideal theory in commutative rings |
| 13B22 | Integral closure of commutative rings and ideals |
References:
| [1] | Arf, C., Une interprétation algébrique de la suite des ordres de multiplicité d’une branche algébrique, Proc. Lond. Math. Soc. (2), 50, 256-287, 1949 · Zbl 0031.07002 |
| [2] | Celikbas, E.; Celikbas, O.; Ciupercă, C.; Endo, N.; Goto, S.; Isobe, R.; Matsuoka, N., On the ubiquity of Arf rings, J. Commut. Algebra, 15, 2, 177-231, 2023 · Zbl 1527.13003 |
| [3] | Endo, N.; Goto, S., Construction of strictly closed rings, Proc. Am. Math. Soc., 150, 1, 119-129, 2022 · Zbl 1478.13003 |
| [4] | Endo, N.; Goto, S.; Isobe, R., Topics on strict closure of rings, Res. Math. Sci., 8, 4, Article 55 pp., 2021, Developments in Commutative Algebra: in honor of Jürgen Herzog on the occasion of his 80th birthday · Zbl 1471.13005 |
| [5] | Isobe, R., Decomposition of integrally closed ideals in Arf rings, J. Commut. Algebra, 15, 3, 335-344, 2023 · Zbl 1537.13006 |
| [6] | Lech, C., A method for constructing bad Noetherian local rings, (Algebra, Algebraic Topology and Their Interactions. Algebra, Algebraic Topology and Their Interactions, Lecture Notes in Mathematics, vol. 1183, 1986, Springer: Springer Berlin, Heidelberg), 241-247 · Zbl 0589.13006 |
| [7] | Lipman, J., Stable ideals and Arf rings, Am. J. Math., 93, 649-685, 1971 · Zbl 0228.13008 |
| [8] | Swanson, I.; Huneke, C., Integral Closure of Ideals, Rings and Modules, 2006, Cambridge University Press · Zbl 1117.13001 |
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