Reductions of ideals in Prüfer rings. (English) Zbl 1518.13002

Let \(R\) be a commutative ring and \(J\subseteq I\) ideals of \(R\). Then \(J\) is a reduction of \(I\) if \(JI^{n}=I^{n+1}\) for some positive integer \(n\), and \(I\) is basic if it has no proper reduction. Two well-known results, due to Hays, assert that an integral domain is Prüfer if and only if every finitely generated ideal is basic, and it is one-dimensional Prüfer if and only if every ideal is basic. The paper under review investigates reductions of ideals in the family of Prüfer rings (with zero-divisors), with the aim to recover and generalize Hays’ results. Two satisfactory results are obtained. The first result asserts that \(w. \dim(R)\leq 1\) if and only if R has the finite basic ideal property; and \(R\) is Prüfer if and only if \(R\) has the finite basic regular ideal property. The second result states that \(w. \dim(R)\leq 1\) and \(\dim(R) \leq 1\) if and only if R has the basic ideal property and if \(R\) is Prüfer with \(\dim(R) \leq 1\), then \(R\) has the basic regular ideal property. The converse is not true in general.


13A15 Ideals and multiplicative ideal theory in commutative rings
13A18 Valuations and their generalizations for commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13F30 Valuation rings
13G05 Integral domains
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[1] J. Abuhlail, M. Jarrar, and S. Kabbaj, “Commutative rings in which every finitely generated ideal is quasi-projective”, J. Pure Appl. Algebra 215:10 (2011), 2504-2511. · Zbl 1226.13014
[2] S. Bazzoni and S. Glaz, “Prüfer rings”, pp. 55-72 in Multiplicative ideal theory in commutative algebra, edited by J. W. Brewer et al., Springer, 2006.
[3] S. Bazzoni and S. Glaz, “Gaussian properties of total rings of quotients”, J. Algebra 310:1 (2007), 180-193. · Zbl 1118.13020
[4] J. G. Boynton, “Prüfer conditions and the total quotient ring”, Comm. Algebra 39:5 (2011), 1624-1630. · Zbl 1241.13017
[5] H. S. Butts and W. Smith, “Prüfer rings”, Math. Z. 95 (1967), 196-211. · Zbl 0153.37003
[6] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, 1956.
[7] L. Fuchs, “Über die Ideale arithmetischer Ringe”, Comment. Math. Helv. 23 (1949), 334-341. · Zbl 0040.30103
[8] S. Glaz, Commutative coherent rings, Lecture Notes in Math. 1371, Springer, 1989. · Zbl 0745.13004
[9] S. Glaz, “Prüfer conditions in rings with zero-divisors”, pp. 272-281 in Arithmetical properties of commutative rings and monoids (Chapel Hill, NC, 2003), edited by S. T. Chapman, Lect. Notes Pure Appl. Math. 241, Chapman & Hall/CRC, Boca Raton, FL, 2005. · Zbl 1107.13023
[10] S. Glaz, “The weak dimensions of Gaussian rings”, Proc. Amer. Math. Soc. 133:9 (2005), 2507-2513. · Zbl 1077.13009
[11] M. Griffin, “Prüfer rings with zero divisors”, J. Reine Angew. Math. 239-240 (1969), 55-67. · Zbl 0185.09801
[12] J. H. Hays, “Reductions of ideals in commutative rings”, Trans. Amer. Math. Soc. 177 (1973), 51-63. · Zbl 0266.13001
[13] J. H. Hays, “Reductions of ideals in Prüfer domains”, Proc. Amer. Math. Soc. 52 (1975), 81-84. · Zbl 0345.13013
[14] C. U. Jensen, “Arithmetical rings”, Acta Math. Acad. Sci. Hungar. 17 (1966), 115-123. · Zbl 0141.03502
[15] S. Kabbaj and A. Mimouni, “Core of ideals in integral domains”, J. Algebra 445 (2016), 327-351. · Zbl 1332.13003
[16] I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, 1970.
[17] D. G. Northcott and D. Rees, “Reductions of ideals in local rings”, Proc. Cambridge Philos. Soc. 50 (1954), 145-158. · Zbl 0057.02601
[18] H. Tsang, Gauss’ lemma, Ph.D. thesis, University of Chicago, 1965, available at https://www.proquest.com/docview/302193002
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