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Normsets of almost Dedekind domains and atomicity. (English) Zbl 1343.13010

An integral domain \(D\) is almost Dedekind if \(D_M\) is a Noetherian valuation domain for all maximal ideals of \(D\). Thus, an almost Dedekind domain \(D\) is Dedekind if and only if it is Noetherian. In this paper, the author introduced a new norm map on almost Dedekind domains as follows: Let \(D\) be an almost Dedekind domain with maximal ideals \(\mathrm{Max}(D)\), for every \(M \in\mathrm{Max}(D)\). It is known that \( D = \bigcap_{M \in\mathrm{Max}(D)} D_{M} \). So one has a map \(V_M: D_M \rightarrow N_0\), where if \( b \in D,V_M (b)=0 \) if \(b\notin M\) and \(V_M (b)>0\) if \(b\in M\). For nonzero \(b\in D\), define the norm of \(b\) to be the net \(N(b) = (V_M (b))_{M \in\mathrm{Max}(D)} \subseteq \prod_{\mathrm{Max}(D)} N_0\). If \(u\) is a unit in \(D\), then \(N(u)\) is the zero net. The author compared and contrasted this new norm map to the traditional Dedekind-Hasse norm. He proved that factoring in an almost Dedekind domain is in one-to-one correspondence to factoring in the new normset, improving upon this results in [J. Coykendall, Proc. Am. Math. Soc. 124, No. 6, 1727–1732 (1996; Zbl 0856.11049)]. In [A. Grams, Proc. Camb. Philos. Soc. 75, 321–329 (1974; Zbl 0287.13002)], an atomic almost Dedekind domain was constructed with a trivial Jacobson radical. The author pursued atomicity in almost Dedekind domains with nonzero Jacobson radicals, showing the usefulness of the new norm he introduced. He also stated theorems with regard to specific classes of almost Dedekind domains and provided a necessary condition for an almost Dedekind domain with nonzero Jacobson radical to be atomic.

MSC:

13A50 Actions of groups on commutative rings; invariant theory
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
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References:

[1] J. Coykendall, Normsets and determination of unique factorization in rings of algebraic integers , Proc. Amer. Math. Soc. 124 (1996), 1727-1732. · Zbl 0856.11049 · doi:10.1090/S0002-9939-96-03261-3
[2] J. Coykendall, D. Dobbs and B. Mullins, On integral domains with no atoms , Comm. Alg. 27 (1999), 5813-5831. · Zbl 0990.13015 · doi:10.1080/00927879908826792
[3] R. Gilmer, Multiplicative Ideal Theory , Queen’s Papers Pure Appl. Math. 90 , Queen’s University Press, Kingston, 1992.
[4] A. Grams, Atomic rings and the ascending chain condition for principal ideals , Proc. Camb. Phil. Soc. 75 (1974), 321-329. · Zbl 0287.13002 · doi:10.1017/S0305004100048532
[5] K.A. Loper, Almost Dedekind domains which are not Dedekind , in Multiplicative ideal theory in commutative algebra ; A tribute to the work of Robert Gilmer , James W. Brewer, Sarah Glaz, William J. Heinzer and Bruce M. Olberding, eds., Springer, New York, 2006.
[6] —-, Sequence domains and integer-valued polynomials , J. Pure Appl. Alg. 119 (1997), 185-210. · Zbl 0960.13005 · doi:10.1016/S0022-4049(96)00025-4
[7] K.A. Loper and T.G. Lucas, Factoring ideals in almost Dedekind domains , J. reine angew. Math. 565 (2003), 61-78. · Zbl 1034.13011 · doi:10.1515/crll.2003.105
[8] B. Olberding, Factorization into radical ideals , in Arithmetical properties of commutative rings and monoids , Lect. Notes Pure Appl. Math. 241 , Chapman & Hall/CRC, Boca Raton, FL, 2005. · Zbl 1091.13002 · doi:10.1201/9781420028249.ch25
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