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Comaximal factorization lattices. (English) Zbl 1494.13022

J. W. Brewer and W. J. Heinzer [Commun. Algebra 30, No. 12, 5999–6010 (2002; Zbl 1049.13002)] studied an integral domain \(D\) in which each nonzero ideal \(A\) can be written as a product \(A = Q_1 \cdots Q_n\), where the \(Q_i\) are pairwise comaximal and have the property that each \(Q_i\) has prime radical (resp., is primary, is prime powers).
Now let \(\mathcal{I}(D)\) be the multiplicative monoid of proper (integral) ideals of an integral domain \(D\). Then \(\mathcal{I}(D)\) is a complete lattice and Brewer and Heinzer’s work is the factorization property of \(\mathcal{I}(D)\) as a lattice. In this paper, the authors extend the results of [loc. cit.] to a more general setting of multiplicative lattices with some additional properties such that \(\mathcal{I}(D)\) is contained in this class of lattices.

MSC:

13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
06F99 Ordered structures
13A05 Divisibility and factorizations in commutative rings

Citations:

Zbl 1049.13002
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References:

[1] Alarcon, F.; Anderson, D. D.; Jayaram, C., Some results on abstract commutative ideal theory, Period. Math. Hung, 30, 1-26 (1995) · Zbl 0822.06015
[2] Anderson, D. D., Abstract commutative ideal theory without chain conditions, Algebra Univ, 6, 131-145 (1976) · Zbl 0355.06022
[3] Anderson, D. D.; Cook, S. J., Two star operations and their induced lattices, Commun. Algebra., 28, 2461-2475 (2000) · Zbl 1043.13001
[4] Anderson, D. D.; Jayaram, C., Principal element lattices, Czech. Math. J, 46, 99-109 (1996) · Zbl 0898.06008
[5] Brewer, J.; Heinzer, W., On decomposing ideals into products of comaximal ideals, Commun. Algebra., 30, 5999-6010 (2002) · Zbl 1049.13002
[6] Dilworth, R., Abstract commutative ideal theory, Pacific J. Math, 12, 481-498 (1962) · Zbl 0111.04104
[7] El Baghdadi, S.; Gabelli, S.; Zafrullah, M., J. Pure Appl. Algebra., 212, 376-393 (2008) · Zbl 1131.13002
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