1,721,039 research outputs found
Star operations on numerical semigroups: antichains and explicit results
Full text linkWe introduce an order on the set of non-divisorial ideals of a numerical semigroup , and link antichains of this order with the star operations on ; subsequently, we use this order to find estimates on the number of star operations on . We then use them to find an asymptotic estimate on the number of nonsymmetric numerical semigroups with or less star operations, and to determine these semigroups explicitly when
Embedding the set of non-divisorial ideals of a numerical semigroup into
Full text linkThe set of the classes of non-divisorial ideals of a numerical semigroup can be endowed with a natural partial order induced by the set of star operations on . We study embeddings of into , specializing on three families of numerical semigroups with radically different behaviour
Star operations on Kunz domains
Full text linkWe study star operations on Kunz domains, a class of analytically irreducible, residually rational domains associated to pseudo-symmetric numerical semigroups, and we use them to refute a conjecture of Houston, Mimouni and Park. We also find an estimate for the number of star operations in a particular case, and a precise counting in a sub-case
Jaffard families and localizations of star operations
Full text linkWe generalize the concept of localization of a star operation to flat overrings; subsequently, we investigate the possibility of representing the set of star operations on as the product of , as ranges in a family of overrings of with special properties. We then apply this method to study the set of star operations on a Pr"ufer domain , in particular the set of stable star operations and the star-class groups of
Calculating the density of solutions of equations related to the Pólya-Ostrowski group through Markov chains
Full text linkMotivated by a problem in the theory of integer-valued polynomials, we investigate the natural density of the solutions of equations of the form , where are fixed integers, are parameters and and are functions related to the -adic valuations of the numbers between 1 and . We show that the number of solutions of this equation in satisfies a recurrence relation, with which we can associate to any pair a stochastic matrix and a Markov chain. Using this interpretation, we calculate the density for the case and for the case , and either or and are coprime
Multiplicative closure operations on ring extensions
No full textLet A⊆B be a ring extension and G be a set of A-submodules of B. We introduce a class of closure operations on G (which we call multiplicative operations on (A,B,G)) that generalizes the classes of star, semistar and semiprime operations. We study how the set Mult(A,B,G) of these closure operations varies when A, B or G vary, and how Mult(A,B,G) behaves under ring homomorphisms. As an application, we show how to reduce the study of star operations on analytically unramified one-dimensional Noetherian domains to the study of closures on finite extensions of Artinian rings
The complete integral closure of a Prüfer domain is a topological property
No full textWe show that the prime spectrum of the complete integral closure D∗ of a Prüfer domain D is completely determined by the Zariski topology on the spectrum Spec(D) of D
The Golomb topology on a Dedekind domain and the group of units of its quotients
Full text linkWe study the Golomb spaces of Dedekind domains with torsion class group. In particular, we show that a homeomorphism between two such spaces sends prime ideals into prime ideals and preserves the P-adic topology on R∖P. Under certain hypothesis, we show that we can associate to a prime ideal P of R a partially ordered set, constructed from some subgroups of the group of units of R/Pn, which is invariant under homeomorphisms, and use this result to show that the unique self-homeomorphisms of the Golomb space of Z are the identity and the multiplication by −1. We also show that the Golomb space of any Dedekind domain contained in the algebraic closure of Q and different from Z is not homeomorphic to the Golomb space of Z
Radicals Of Principal Ideals And The Class Group Of A Dedekind Domain
No full textFor a Dedekind domain D, let P.D/ be the set of ideals of D that are the radical of a principal ideal. We show that, if D and D0 are Dedekind domains and there is an order isomorphism between P.D/ and P.D0/, then the rank of the class groups of D and D0 is the same
Almost Dedekind domains without radical factorization
No full textWe 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 = Max(R) of an almost Dedekind domain R, interpreting its (fractional) ideals as maps from M to Z, and looking at the continuity of these maps when M is endowed with the inverse topology and Z with the discrete topology. We generalize the concept of critical ideals by introducing a well-ordered chain of closed subsets of 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 is scattered and, in particular, the almost Dedekind domains such that M is countable. We show that for this class of rings the group Inv (R) is free by expressing it as a direct sum of groups of continuous maps, and that, for every length function l on R and every ideal I of R, the length of R / I is equal to the length of R / rad(I)
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