Related papers: Ideals in the ring of Colombeau generalized number…
A symmetric chain of ideals is a rule that assigns to each finite set $S$ an ideal $I_S$ in the polynomial ring $\mathbb{C}[x_i]_{i \in S}$ such that if $\phi \colon S \to T$ is an embedding of finite sets then the induced homomorphism…
The notion of 2-AGL ring in dimension one which is a natural generalization of almost Gorenstein local ring is posed in terms of the rank of Sally modules of canonical ideals. The basic theory is developed, investigating also the case where…
In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an N-graded ring generated by…
This article is the natural continuation of the paper: Mukhammadiev A.~et al Supremum, infimum and hyperlimits of Colombeau generalized numbers in this journal. Since the ring $\tilde{R}$ of Robinson-Colombeau is non-Archimedean, a…
The aim of this work is to study sets of values of fractional ideals of rings of algebroid curves and explore more deeply the symmetry that exists among sets of values of dual pairs of ideals when the ring is Gorenstein. We also express the…
The theory of ternary $\Gamma$-semirings extends classical ring and semiring frameworks by introducing a ternary product controlled by a parameter set $\Gamma$. Building on the foundational axioms recently established by Rao, Rani, and…
We develop the theory of central ideals on commutative rings. We introduce and study the central seminormalization of a ring in another one. This seminormalization is related to the theory of regulous functions on real algebraic varieties.…
We construct a local Cohen-Macaulay ring $R$ with a prime ideal $\mathfrak{p}\in\spec(R)$ such that $R$ satisfies the uniform Auslander condition (UAC), but the localization $R_{\mathfrak{p}}$ does not satisfy Auslander's condition (AC).…
Let $C \subset {\bf N}^d$ be an affine semigroup, and $R=K[C]$ its semigroup ring. This paper is a collection of various results on "$C$-graded" $R$-modules, especially, monomial ideals. For example, we show the following: If $R$ is normal…
We obtain various characterizations of commutative Noetherian local rings $(R, \fm)$ in terms of homological dimensions of certain finitely generated modules. For example, we establish that $R$ is Gorenstein if the Gorenstein injective…
Over the past few years there has been considerable progress in the structural understanding of special Colombeau algebras. We present some of the main trends in this development: non-smooth differential geometry, locally convex theory of…
In this article we study some algebraic aspects of multicomplex numbers $\mathbb M_n$. For $n\geq 2$ a canonical representation is defined in terms of the multiplication of $n-1$ idempotent elements. This representation facilitates…
We show that spaces of Colombeau generalized functions with smooth parameter dependence are isomorphic to those with continuous parametrization. Based on this result we initiate a systematic study of algebraic properties of the ring…
The main goal of this article is to introduce BL-rings, i.e., commutative rings whose lattices of ideals can be equipped with a structure of BL-algebra. We obtain a description of such rings, and study the connections between the new class…
The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra and in algebraic geometry. In this paper we give a complete algebraic structure of the Sally module of integrally closed ideals $I$ in a…
For a Cohen-Macaulay local ring $(R,\mathfrak{m})$ with canonical module, we study how relations between $\text{index}(R)$ and $\text{g}\ell\ell(R)$ and between $\text{index}(R)$ and $e(R)$ are preserved when factoring out regular sequences…
Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining…
In this paper we study Ulrich ideals of and Ulrich modules over Cohen--Macaulay local rings from various points of view. We determine the structure of minimal free resolutions of Ulrich modules and their associated graded modules, and…
The third named author and P\'{e}rez proved that under certain conditions the test ideal of a module closure agrees with the trace ideal of the module closure. We use this fact to compute the test ideals of various rings with respect to the…
In this paper we generalize the theorem of Ein-Lazarsfeld-Smith (concerning the behavior of symbolic powers of prime ideals in regular rings finitely generated over a field of characteristic 0) to arbitrary regular rings containing a field.…