On Krull domains. (English) Zbl 0777.13011

One aim of this article is to provide for Krull domains a star-operation analogue of the following result: An integral domain \(D\) is a Dedekind domain if and only if each nonzero ideal \(A\) of \(D\) is strongly two generated. A nonzero ideal \(A\) of an integral domain \(D\) is called strongly two generated if for each \(x\in A\backslash\{0\}\) there is \(y\in A\) such that \(A=xD+yD\). D. C. Lantz and M. R. Martin showed in Commun. Algebra 16, No. 9, 1759-1777 (1988; Zbl 0655.13015) that a strongly two generated ideal is invertible. Following this lead we define a strongly \(*\)-type 2 ideal, for a star-operation \(*\), as a nonzero ideal \(A\) such that for each \(x\in A\backslash\{0\}\), there is \(y\in A^*\) such that \((x,y)^*=A^*\). Then in section 1 we characterize Krull domains in terms of strongly \(*\)-type 2 ideals. Recently there has been considerable activity in characterizing a Krull domain in terms of the \(*\)-invertibility of some or all fractional ideals of \(D\). These results are interesting in that they indicate that most of the characterizations of Dedekind domains have \(*\)-operation analogues for Krull domains.
In section 2 we continue this line of investigation by coordinating some of the recent results with some new characterizations of Krull domains in terms of \(*\)-invertibility.


13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13A15 Ideals and multiplicative ideal theory in commutative rings
13E15 Commutative rings and modules of finite generation or presentation; number of generators


Zbl 0655.13015
Full Text: DOI


[1] D. D. Anderson, J. Mott andM. Zafrullah, Some quotient based statements in multiplicative ideal theory. Boll. Mat. Ital. (7)3-B, 455-476 (1989). · Zbl 0676.13003
[2] D. D. Anderson andB. G. Kang, Pseudo-Dedekind domains and divisorial ideals inR[X] T . J. Algebra122, 323-336 (1989). · Zbl 0698.13004
[3] V. Barucci, On a class of Mori domains. Comm. Algebra11, 1989-2001 (1983). · Zbl 0518.13012
[4] V. Barucci andS. Gabelli, How far is a Mori domain from being a Krull domain? J. Pure Appl. Algebra45, 101-112 (1987). · Zbl 0623.13008
[5] N.Bourbaki, Elements of Mathematics, Commutative Algebra. Reading 1972. · Zbl 0279.13001
[6] J. Brewer andW. Heinzer, Associated primes of principal ideals. Duke Math. J.41, 1-7 (1974). · Zbl 0284.13001
[7] D. Dobbs, E. Houston, T. Lucas andM. Zafrullah,t-linked overrings and Prüferv-multiplication domains. Comm. Algebra (11)17, 2835-2852 (1989). · Zbl 0691.13015
[8] R.Fossum, The Divisor Class Group of a Krull Domain. Berlin-Heidelberg-New York 1973. · Zbl 0256.13001
[9] R.Gilmer, Multiplicative Ideal Theory. New York 1972. · Zbl 0248.13001
[10] M. Griffin, Some results onv-multiplication rings. Canad. J. Math.19, 710-722 (1967). · Zbl 0148.26701
[11] W. Heinzer, Integral domains in which each nonzero ideal is divisorial. Mathematika15, 164-170 (1968). · Zbl 0169.05404
[12] E. Houston andM. Zafrullah, Integral domains in which eacht-ideal is divisorial. Mich. Math. J.35, 291-300 (1988). · Zbl 0675.13001
[13] E. Houston andM. Zafrullah, Ont-invertibility II. Comm. Algebra (8)17, 1955-1969 (1989). · Zbl 0717.13002
[14] P.Jaffard, Les systems d’Ideaux. Paris 1960. · Zbl 0101.27502
[15] B. G. Kang, On the converse of a well-known fact about Krull domains. J. Algebra124, 284-299 (1989). · Zbl 0694.13011
[16] I.Kaplansky, Commutative Rings. Boston 1970.
[17] D. Lantz andM. Martin, Strongly two-generated ideals. Comm. Algebra16, 1759-1777 (1988). · Zbl 0655.13015
[18] S. Malik, J. Mott andM. Zafrullah, Ont-invertibility. Comm. Algebra16, 149-170 (1988). · Zbl 0638.13002
[19] J.Mott, B.Nashier and M.Zafrullah, Contents of polynomials and invertibility. Comm. Algebra, to appear. · Zbl 0705.13005
[20] J. Mott andM. Zafrullah, Unruly Hilbert domains. Canad. Math. Bull. (1)33, 106-109 (1990). · Zbl 0757.13010
[21] T. Nishimura, Unique factorization of ideals in the sense of quasi-equality. J. Math. Kyoto Univ.3, 115-125 (1963). · Zbl 0129.02103
[22] J. Querre, Sur une propriete des anneaux de Krull. Bull. Sci. Math. (2)95, 341-354 (1971).
[23] J. Querre, Ideaux divisorials d’un anneau de polynomes. J. Algebra64, 270-284 (1980). · Zbl 0441.13012
[24] U.Storch, Fastfaktorielle Ringe. Schriftreihe Math. Inst. Univ. Münster36 (1967).
[25] M. Zafrullah, Generalized Dedekind domains. Mathematika33, 285-295 (1986). · Zbl 0613.13001
[26] M. Zafrullah, Ascending chain condition and star operations. Comm. Algebra (6)17, 1523-1533 (1989). · Zbl 0691.13013
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