Almost canonical ideals and GAS numerical semigroups

Abstract

We propose the notion of GAS numerical semigroup which generalizes both almost symmetric and 2-AGL numerical semigroups. Moreover, we introduce the concept of almost canonical ideal which generalizes the notion of canonical ideal in the same way almost symmetric numerical semigroups generalize symmetric ones. We prove that a numerical semigroup with maximal ideal M and multiplicity e is GAS if and only if M − e is an almost canonical ideal of M − M . This generalizes a result of Barucci about almost symmetric semigroups and a theorem of Chau, Goto, Kumashiro, and Matsuoka about 2-AGL semigroups. We also study the transfer of the GAS property from a numerical semigroup to its gluing, numerical duplication and dilatation.

When is $\mathfrak{m}:\mathfrak{m}$ an almost Gorenstein ring?

“…Some of these results and definitions generalize corresponding facts and concepts given in the numerical semigroups context ( [9]). Let us notice that this generalization is not straightforward, since we have to deal with two problems that do not appear looking at numerical semigroups: the fact that B could not be local anymore and the non-residually rational case, i.e.…”

“…In [9] a relative ideal E of S is said to be almost canonical if E − M = K(S) ∪ {F(S)}. This notion was introduced in order to generalize the concept of almost symmetric numerical semigroup, since S is almost symmetric if and only if S − M = K(S) ∪ {F(S)}.…”

“…In this section we introduce a new class of rings which includes both almost Gorenstein and 2-AGL rings; moreover, this will be a generalization of the notion of GAS numerical semigroup given in [9]. First of all we notice that in a GAS numerical semigroup S we always have 2M ⊆ (S − K(S)), see [9, Proposition 3.5], so we assume that m 2 ⊆ (R : ω R ).…”

“…Even though the very definition of GAS numerical semigroup is related to the pseudo-Frobenius numbers of the semigroup, that are in turn connected to the generators of the socle of the associated ring, in [9,Proposition 3.5] it is proved that a numerical semigroup S is GAS if and only if x − y is not in the maximal ideal of M − M for every x, y ∈ M \ (S − K(S)). Moreover, excluding the Gorenstein case, it is showed that this is also equivalent to say that 2M ⊆ S − K(S) ⊆ M and the maximal ideal of M − M is contained in (S − K(S)) − M .…”