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• W3208658608 abstract "Abstract We study almost Dedekind domains with respect to the failure of ideals to have radical factorization, that is, we study how to measure how far an almost Dedekind domain is from being an SP-domain. To do so, we consider the maximal space <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi mathvariant=script>ℳ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>Max</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>R</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {mathcal{M}=operatorname{Max}(R)} of an almost Dedekind domain R , interpreting its (fractional) ideals as maps from <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>ℳ</m:mi> </m:math> {mathcal{M}} to <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>ℤ</m:mi> </m:math> {mathbb{Z}} , and looking at the continuity of these maps when <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>ℳ</m:mi> </m:math> {mathcal{M}} is endowed with the inverse topology and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>ℤ</m:mi> </m:math> {mathbb{Z}} with the discrete topology. We generalize the concept of critical ideals by introducing a well-ordered chain of closed subsets of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>ℳ</m:mi> </m:math> {mathcal{M}} (of which the set of critical ideals is the first step) and use it to define the class of SP-scattered domains , which includes the almost Dedekind domains such that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>ℳ</m:mi> </m:math> {mathcal{M}} is scattered and, in particular, the almost Dedekind domains such that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=script>ℳ</m:mi> </m:math> {mathcal{M}} is countable. We show that for this class of rings the group <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>Inv</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>R</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {mathrm{Inv}(R)} is free by expressing it as a direct sum of groups of continuous maps, and that, for every length function <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=normal>ℓ</m:mi> </m:math> {ell} on R and every ideal I of R , the length of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>R</m:mi> <m:mo>/</m:mo> <m:mi>I</m:mi> </m:mrow> </m:math> {R/I} is equal to the length of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>R</m:mi> <m:mo>/</m:mo> <m:mrow> <m:mi>rad</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>I</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {R/{operatorname{rad}(I)}} ." @default.
• W3208658608 created "2021-11-08" @default.
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• W3208658608 date "2023-01-30" @default.
• W3208658608 modified "2023-09-25" @default.
• W3208658608 title "Almost Dedekind domains without radical factorization" @default.
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