Ulrich ideals and 2-AGL rings

Abstract

The notion of 2-almost Gorenstein local ring (2-AGL ring for short) is a generalization of the notion of almost Gorenstein local ring from the point of view of Sally modules of canonical ideals. In this paper, for further developments of the theory, we discuss three different topics on 2-AGL rings. The first one is to clarify the structure of minimal presentations of canonical ideals, and the second one is the study of the question of when certain fiber products, so called amalgamated duplications are 2-AGL rings. We also explore Ulrich ideals in 2-AGL rings, mainly two-generated ones.

Almost canonical ideals and GAS numerical semigroups

“…These rings generalize almost Gorenstein ones that are obtained when either n = 0, in which case the ring is Gorenstein, or n = 1. In particular, in [6] it is studied the case of the 2-AGL rings, that are closer to be almost Gorenstein, see also [11].…”

“…Indeed a ring is almost Gorenstein if and only if it is either 1-AGL or 0-AGL, with 0-AGL equivalent to be Gorenstein. In this respect 2-AGL rings are near to be almost Gorenstein and for this reason their properties have been deepened in [6,11]. In [6] it is also studied the numerical semigroup case, where 2-AGL numerical semigroups are close to be almost symmetric.…”

Ulrich ideals in numerical semigroup rings of small multiplicity

“…In a one-dimensional non-Gorenstein almost Gorenstein local ring, the only possible Ulrich ideal is the maximal ideal ([10, Theorem 2.14]). In [7] the authors explored the ubiquity of Ulrich ideals in a 2-AGL rings (one of the generalizations of Gorenstein local rings of dimension one), and showed that the existence of two-generated Ulrich ideals provides a rather strong restriction on the structure of the base local rings ( [7,Theorem 4.7]). Nevertheless, even for the one-dimensional Cohen-Macaulay local rings, in general we lack an explicit and physical list of Ulrich ideals contained inside those rings, which possibly prevents further developments of the study of Ulrich ideals.…”

“…Nevertheless, even for the one-dimensional Cohen-Macaulay local rings, in general we lack an explicit and physical list of Ulrich ideals contained inside those rings, which possibly prevents further developments of the study of Ulrich ideals. In order to supply the lack, continuing the work [7], the present research particularly focus on and investigate the question of how many and how ample two-generated Ulrich ideals are contained in a given numerical semigroup ring, which is a prototype of Cohen-Macaulay local rings of dimension one. As we shall show in the following, although the task is rather tough and the statements of the results are seemingly complicated, we are able to describe all the Ulrich ideals in certain specific numerical semigroup rings.…”

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

“…More precisely, every such ring is n-AGL for some n and Gorenstein rings correspond to the 0-AGL rings, whereas a ring is 1-AGL if and only if it is almost Gorenstein ring but not Gorenstein. Therefore, in this perspective 2-AGL rings are the ones closer to be almost Gorenstein and for this reason their properties have been more studied, see, e.g., [11]. In particular, the notion of 2-AGL gives a partial answer to a natural question that arise studying the endomorphism algebra B: can we characterize when it is almost Gorenstein?…”