Related papers: Burch ideals and Burch rings
Fiber cones of 0-dimensional ideals with almost minimal multiplicity in Cohen-Macaulay local rings are studied. Ratliff-Rush closure of filtration of ideals with respect to another ideal is introduced. This is used to find a bound on the…
This is an introduction to rings and fields, written for a quarter-long undergraduate course. It includes the basic properties of ideals, modules, algebras and polynomials, the constructions of ring extensions and finite fields, some…
We compute the arithmetic ranks of the defining ideals of homogeneous coordinate rings of certain Segre products arising from elliptic curves. The cohomological dimension of these ideals varies with the characteristic of the field, though…
To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By…
We consider a classification problem of ideals by codimension in case rings are the local rings of irreducible curve singularities. In this paper, we introduce a systematic method to solve this problem.
We study a topological generalization of ideal co-maximality in topological rings and present some of its properties, including a generalization of the Chinese remainder theorem. Using the hyperspace uniformity, we prove a stronger version…
We investigate ideal-semisimple and congruence-semisimple semirings. We give several new characterizations of such semirings using e-projective and e-injective semimodules. We extend several characterizations of semisimple rings to (not…
The aim of this survey is to discuss invariants of Cohen-Macaulay local rings that admit a canonical module. Attached to each such ring R with a canonical ideal C, there are integers--the type of R, the reduction number of C--that provide…
We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the…
We study exceptional minuscule Schubert varieties and provide the defining equations of the defining ideals of their intersection with the big open cell. We also provide the resolutions of these ideals and characterize some of them in terms…
In this paper, we introduce the notions of tight closure of ideals on Witt rings and quasi-tightly closedness of system of parameters. By using the notions, we obtain a characterization of quasi-$F$-rationality. Furthermore, we study the…
We discuss algebraic and homological properties of binomial edge ideals associated to graphs which are obtained by gluing of subgraphs and the formation of cones.
Let $b$ be a fractional ideal of a one-dimensional Cohen-Macaulay local ring $O$ containing a perfect field $k$. This paper is devoted to the study some $O$-modules associated with $b$. In addition, different motivic Poincar\'e series are…
We generalize the notion of K\"ulshammer ideals to the setting of a graded category. This allows us to define and study some properties of K\"ulshammer type ideals in the graded center of a triangulated category and in the Hochschild…
In this paper, we introduce the concept of $S$-B\'ezout ring, as a generalization of B\'ezout ring. We investigate the relationships between $S$-B\'ezout and other related classes of rings. We establish some characterizations of…
This paper investigates the relationship between multiplicities and the degree sequence of ideals in graded algebras, gives multiplicity equations of graded rings via the degree sequence of ideals, and characterizes mixed multiplicities and…
The aim of this note is to study the class of one dimensional Cohen-Macaulay local rings, $(R, \mathfrak{m})$ say, possessing a canonical ideal $K$ which is a reduction of $\mathfrak{m}$. We call $R$ to have canonical reduction $K$. We show…
Trusses, defined as sets with a suitable ternary and a binary operations, connected by the distributive laws, are studied from a ring and module theory point of view. The notions of ideals and paragons in trusses are introduced and several…
It is shown that a module is sequentially Cohen-Macaulay if and only if the index of reducibility for distinguished parameter ideals are eventually constant with special value. As corollaries to the main theorem we given to characterize the…
We describe the universal Groebner basis of the ideal of maximal minors and the ideal of $2$-minors of a multigraded matrix of linear forms. Our results imply that the ideals are radical and provide bounds on the regularity. In particular,…