Mathematics

We investigate the nearly Gorenstein property among d -dimensional cyclic quotient singularities k[[ x 1 ,…, x d ] ] G , where k is an algebraically closed field and G⊆GL(d,k) is a finite small cyclic group whose order is invertible in k . We prove a necessary and sufficient condition to be nearly Gorenstein that also allows us to find several new classes of such rings.

We extend some results on almost Gorenstein affine monomial curves to the nearly Gorenstein case. In particular, we prove that the Cohen-Macaulay type of a nearly Gorenstein monomial curve in A 4 is at most 3 , answering a question of Stamate in this particular case. Moreover, we prove that, if C is a nearly Gorenstein affine monomial curve which is not Gorenstein and n 1 ,…, n ν are the minimal generators of the associated numerical semigroup, the elements of { n 1 ,…, n i ˆ ,…, n ν } are relatively coprime for every i .

A primary ideal in a polynomial ring can be described by the variety it defines and a finite set of Noetherian operators, which are differential operators with polynomial coefficients. We implement both symbolic and numerical algorithms to produce such a description in various scenarios as well as routines for studying affine schemes through the prism of Noetherian operators and Macaulay dual spaces.

Let P be a finite poset, K a field, and O(P) (resp. C(P) ) the order (resp. chain) polytope of P . We study the non-Gorenstein locus of E K [O(P)] (resp. E K [C(P)] ), the Ehrhart ring of O(P) (resp. C(P) ) over K , which are each normal toric rings associated P . In particular, we show that the dimension of non-Gorenstein loci of E K [O(P)] and E K [C(P)] are the same. Further, we show that E K [C(P)] is nearly Gorenstein if and only if P is the disjoint union of pure posets P 1 ,…, P s with |rank P i −rank P j |≤1 for any i and j .

Let (A,m) be an excellent two-dimensional normal local domain. The geometric genus p g (A) is an important geometric invariant of A . A rational singularity is characterized by p g (A)=0 and the integrally closed m -primary ideals of A are normal and well described by Cutkosky and Lipman. Later, Okuma, Watanabe and Yoshida characterized rational singularities through the p g -ideals. In this paper we define the elliptic and the strongly elliptic ideals of A with the aim to characterize elliptic and strongly elliptic singularities, according to the definitions given by Wagreich and by Yau. A strongly elliptic singularity can be described by p g (A)=1. We characterize a strongly elliptic singularity in terms of the normal Hilbert coefficients of the integrally closed m -primary ideals of A . Unlike p g -ideals, elliptic and strongly elliptic ideals are not necessarily normal and we give necessary and sufficient conditions for being normal. In the last section we discuss the existence (and the effective construction) of strongly elliptic ideals in any 2 -dimensional normal local ring.

Let R be a commutative ring with nonzero identity. Let I(R) be the set of all ideals of R and let δ:I(R)?�I(R) be a function. Then δ is called an expansion function of ideals of R if whenever L,I,J are ideals of R with J?�I , we have L?��?L) and δ(J)?��?I) . Let δ be an expansion function of ideals of R . In this paper, we introduce and investigate a new class of ideals that is closely related to the class of δ -primary ideals. A proper ideal I of R is said to be a 1 -absorbing δ -primary ideal if whenever nonunit elements a,b,c?�R and abc?�I , then ab?�I or c?��?I). Moreover, we give some basic properties of this class of ideals and we study the 1 -absorbing δ -primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.

In this paper, we introduce ϕ -1-absorbing prime ideals in commutative rings. Let R be a commutative ring with a nonzero identity 1≠0 and ϕ:I(R)→I(R)∪{∅} be a function where I(R) is the set of all ideals of R . A proper ideal I of R is called a ϕ -1-absorbing prime ideal if for each nonunits x,y,z∈R with xyz∈I−ϕ(I) , then either xy∈I or z∈I . In addition to give many properties and characterizations of ϕ -1-absorbing prime ideals, we also determine rings in which every proper ideal is ϕ -1-absorbing prime.

Let R be a commutative ring with identity. In this paper, we introduce the concept 1-absrbing primary ideal of R.

In this study, we aim to introduce the concept of a 1-absorbing prime submodule of an unital module over a commutative ring with a non-zero identity. Let M be an R-module and N be a proper submodule of M. For all non-unit elements a, b in R and m in M if abm in N, either ab in (N : M) or m in N, then N is called 1-absorbing prime submodule of M. We show that the new concept is a generalization of prime submodules at the same time it is a kind of special 2-absorbing submodule. In addition to some properties of a 1-absorbing prime submodule, we obtain a characterization of it in a multiplication module.

We introduce Artinian Gorenstein algebras defined by the face posets of regular polyhedra. We consider the strong Lefschetz property and Hodge--Riemann relation for the algebras. We show the strong Lefschetz property of the algebras for all Platonic solids. On the other hand, for some Platonic solids, we show that the algebras do not satisfy the Hodge--Riemann relation with respect to some strong Lefschetz elements.