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Dilatations of numerical semigroups
No full textThis paper is focused on numerical semigroups and presents a simple construction, that we call “dilatation”, which, from a starting semigroup S, permits to get an infinite family of semigroups which share several properties with S. The invariants of each semigroup T of this family are given in terms of the corresponding invariants of S and both the Apéry set and the minimal generators of T are described. We also study three properties that are close to the Gorenstein property of the associated semigroup ring: almost Gorenstein, 2-AGL, and nearly Gorenstein properties. More precisely, we prove that S satisfies one of these properties if and only if each dilatation of S satisfies the corresponding one
Families of Gorenstein and almost Gorenstein rings
No full textStarting with a commutative ring R and an ideal I, it is possible to define a family of rings R(I)_{a,b}, with a, b∈R, as quotients of the Rees algebra ⊕_{n≥0} I^n t^n; among the rings appearing in this family we find Nagata’s idealization and amalgamated duplication. Many properties of these rings depend only on R and I and not on a, b; in this paper we show that the Gorenstein and the almost Gorenstein properties are independent of a, b. More precisely, we characterize when the rings in the family are Gorenstein, complete intersection, or almost Gorenstein and we find a formula for the type
Maximality properties in numerical semigroups with applications to one- dimensional analytically irreducible local domains
No full text
Dilatations of numerical semigroups
No full textThis paper is focused on numerical semigroups and presents a simple construction, that we call "dilatation", which, from a starting semigroup S, permits to get an infinite family of semigroups which share several properties with S. The invariants of each semigroup T of this family are given in terms of the corresponding invariants of S and both the Apéry set and the minimal generators of T are described. We also study three properties that are close to the Gorenstein property of the associated semigroup ring: almost Gorenstein, 2-AGL, and nearly Gorenstein properties. More precisely, we prove that S satisfies one of these properties if and only if each dilatation of S satisfies the corresponding one
A family of quotients of the Rees Algebra
No full textA family of quotient rings of the Rees algebra associated to a com-
mutative ring is studied. This family generalizes both the classical
concept of idealization by Nagata and a more recent concept, the
amalgamated duplication of a ring. It is shown that several properties
of the rings of this family do not depend on the particular member
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