Let K be a number field and p be a prime number. When we factor (p) = pOK into prime ideals, say (p) = p1 1 · · · p eg g , we refer to the data of the ei’s, the exponents fi in the norms Npi = p fi , and g (the number of pi’s), as the “shape” of the factorization of (p). (What we are leaving out of this is explicit information about generators for the ideals pi.) Similarly, if a monic polynomia...

This paper introduces two new representations for real-parameter spaces—the Dedekind and Isodedekind representations. Point mutation and uniform crossover—in their generalised, representation-independent form—are shown, when instantiated with respect to these representations, to give rise to familiar operators for continuous domains, such as gaussian mutation, blend crossover and line recombina...

Various contributors to recent philosophy of mathematics have taken Richard Dedekind to be the founder of structuralism in mathematics. In this paper I examine whether Dedekind did, in fact, hold structuralist views and, insofar as that is the case, how they relate to the main contemporary variants. In addition, I argue that his writings contain philosophical insights that are worth reexamining...

In 1977 Hartwig and Luh asked whether an element a in a Dedekind-finite ring R satisfying aR = a2R also satisfies Ra = Ra2. In this paper, we answer this question in the negative. We also prove that if a is an element of a Dedekind-finite exchange ring R and aR = a2R, then Ra = Ra2. This gives an easier proof of Dischinger’s theorem that left strongly π-regular rings are right strongly π-regula...

Abstract Let R be an integral domain with $qf(R)=K$ , and let $F(R)$ the set of nonzero fractional ideals . Call a dually compact (DCD) if, for each $I\in F(R)$ ideal $I_{v}=(I^{-1})^{-1}$ is finite intersection principal ideals. We characterize DCDs show that class properly contains various classes domains, such as Noetherian, Mori, Krull domains. In addition, we Schreier DCD greatest common d...