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\(\ast \)-finite ideals contained in infinitely many maximal \(\ast _{s}\)-ideals. (English) Zbl 1238.13004

The authors give a new characterization of \(*\)-finite ideals that are contained in infinitely many \(*_{f}\)-maximal ideals. That is, let \(R\) be an integral domain, \(*\) a star operation on \(R\), \(*_{f}\) the star operation of finite type associated to \(*\) and \(\Gamma\) a set of proper \(*\)-ideals of finite type of \(R\) such that every proper \(*\)-finite \(*\) ideal of \(R\) is contained in some element of \(\Gamma\). Let \(A\) be a nonzero finitely generated ideal of \(R\) with \(A^{*}\not =R\). Then \(A\) is contained in an infinite number of \(*_{f}\)-maximal ideals of \(R\) if and only if there exists an infinite family of mutually \(*_{f}\)-comaximal ideals in \(\Gamma\) containing \(A\).

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
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References:

[1] Dumitrescu, T., Zafrullah, M.: Characterizing domains of finite character. J. Pure Appl. Algebra 214, 2087–2091 (2010) · Zbl 1202.13001 · doi:10.1016/j.jpaa.2010.02.014
[2] Gilmer, R.: Multiplicative Ideal Theory. Dekker, New York (1972) · Zbl 0248.13001
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