Abstract
In this paper we present and study the ideal duplication, a new construction within the class of the relative ideals of a numerical semigroup S, that, under specific assumptions, produces a relative ideal of the numerical duplication Sbowtie ^b E. We prove that every relative ideal of the numerical duplication can be uniquely written as the ideal duplication of two relative ideals of S; this allows us to better understand how the basic operations of the class of the relative ideals of Sbowtie ^b E work. In particular, we characterize the ideals E such that Sbowtie ^b E is nearly Gorenstein.
Highlights
The numerical duplication is a construction introduced in [7] that, starting with a numerical semigroup S and a semigroup ideal E ⊂ S, produces a new numerical semigroup, denoted by S b E
The origin of this construction is connected to ring theory; more precisely, in [2] it is studied a family of quadratic quotients of the Rees algebra of a ring R with respect to an ideal I, with the aim of giving a unified approach for Nagata’s idealization and amalgamated duplication; when the original ring R is a numerical semigroup ring or an algebroid branch, so it has a value semigroup S = v(R), particular members of the family are again numerical semigroup rings whose value semigroup is a numerical duplication of S
In this paper we want to provide a systematical study of relative ideals of a numerical duplication; to this aim we introduce a similar construction for relative ideals, that we call ideal duplication: given a numerical semigroup S, an odd number b ∈ S and an ideal E, the ideal duplication starting from two relative ideals E1, E2 of S, under specific assumptions, produces a relative ideal E1 b E2 of the numerical duplication S b E
Summary
The numerical duplication is a construction introduced in [7] that, starting with a numerical semigroup S and a semigroup ideal E ⊂ S, produces a new numerical semigroup, denoted by S b E (where b is any odd integer belonging to S). 4, using the ideal duplication, we are able to describe the trace of the numerical duplication (see Theorem 4.2) and, from this, it follows a characterization of the nearly Gorenstein property (see Corollary 4.3); we apply this result to study the nearly Gorensteinness for numerical duplications obtained by some particular classes of ideals, like e.g. integrally closed ideals (see Theorem 4.11). We give a description of the pseudo-Frobenius numbers of the numerical duplication (see Theorem 4.12) which produces a new characterization for S b E to be almost symmetric (see Corollary 4.13)
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