Relative Igusa-Todorov Functions and Relative Homological Dimensions

We develope the theory of \({\mathcal {E}}\)-relative Igusa-Todorov functions in an exact I T-context \(({\mathcal {C}},{\mathcal {E}})\) (see Definition 2.1). In the case when \({\mathcal {C}}={\text {mod}}\, ({\Lambda })\) is the category of finitely generated left Λ-modules, for an artin algebra Λ, and \({\mathcal {E}}\) is the class of all exact sequences in \({\mathcal {C}},\) we recover the usual Igusa-Todorov functions, Igusa K. and Todorov G. (2005). We use the setting of the exact structures and the Auslander-Solberg relative homological theory to generalise the original Igusa-Todorov’s results. Furthermore, we introduce the \({\mathcal {E}}\)-relative Igusa-Todorov dimension and also we obtain relationships with the relative global and relative finitistic dimensions and the Gorenstein homological dimensions.     

Completely Prime Ideals in Multiplicatively Idempotent Semirings

Mathematical Notes - Structure properties of multiplicatively idempotent semirings are considered. Basic results concerning completely prime ideals in multiplicatively idempotent semirings are...     

幂等元中心的置换理想(英文)

证明了幂等元中心的置换理想有稳定秩1;进一步地,证明了幂等元中心的置换理想具有一类部分一般比较性.     

Essential Ideals in Rings of Measurable Functions

In this article, we study essential ideals, socle, and some related ideals of rings of real valued measurable functions. We also study the Goldie dimension of rings of measurable functions.     

Closed Ideals of Holomorphic Functions with Two Generators

Mathematical Notes - For a wide class of algebras P of holomorphic functions on a domain in ?, sufficient conditions are obtained under which a closed ideal I ? P is 2-generated.     

Hilbert Functions, Residual Intersections, and Residually S2 Ideals

Let R be a homogeneous ring over an infinite field, IR a homogeneous ideal, and I an ideal generated by s forms of degrees d ₁,...,d _s so that codim( :I)s. We give broad conditions for when the Hilbert function of R/ or of R/( :I) is determined by I and the degrees d ₁,...,d _s . These conditions are expressed in terms of residual intersections of I, culminating in the notion of residually S ₂ ideals. We prove that the residually S ₂ property is implied by the vanishing of certain Ext modules and deduce that generic projections tend to produce ideals with this property.     

解析函数的加权代数中的闭理想问题

该文讨论了单位圆盘上解析函数的加权代数犃犘中的闭理想问题,并且利用加权系给出了一个闭理想成为犃犘中的补子空间的判断准则     

Characterization of Generating Ideals in Some Rings of Entire Functions

Let E be a ring of entire functions on with the operation of pointwise multiplication, and let f ₁,...,f _m be a set of nonzero elements in E. The ideal E(f ₁,...,f _m ) in E with generators f ₁,...,f _m is said to be generating if E(f ₁,...,f _m ) = E. The generating ideals in rings of entire functions on determined by the growth of their maximum moduli are characterized in terms of the distribution of the zero sets of their generators. Under the additional condition of rapid variation of the weight sequences determining the ring, criteria for generating ideals are established; they are stated in terms of d(z) max _{1 j m} d _j (z), where d _j (z) is the distance from a point to the zero set of f _j for 1 j m. It is shown that, in rings of entire functions of finite or minimal type with respect to a given order, a similar characterization (i.e., in terms of d(z)) cannot be given.     

Idempotent uninorms

Uninorms are an important generalization of t-norms and t-conorms, having a neutral element lying anywhere in the unit interval. Two broad classes of idempotent uninorms are fully characterized: the class of left-continuous ones and the class of right-continuous ones. In particular, the important subclasses of conjunctive left-continuous idempotent uninorms and of disjunctive right-continuous idempotent uninorms are characterized by means of super-involutive and sub-involutive decreasing unary operators. As a consequence, it is shown that any involutive negator gives rise to a conjunctive left-continuous idempotent uninorm and to a disjunctive right-continuous idempotent uninorm.     

On Generators in Ideals of Entire Functions of Finite Order and Type on the Plane

We study ideals in the space of entire functions of finite order and type on the plane. We obtain necessary and sufficient conditions for the ideal of functions vanishing at the common zeros of a system of N functions to be generated by the latter.     

Soft Ideals and Arithmetic Mean Ideals

This article investigates the soft-interior (se) and the soft-cover (sc) of operator ideals. These operations, and especially the first one, have been widely used before, but making their role explicit and analyzing their interplay with the arithmetic mean operations is essential for the study in [10] of the multiplicity of traces. Many classical ideals are ‘soft’, i.e., coincide with their soft interior or with their soft cover, and many ideal constructions yield soft ideals. Arithmetic mean (am) operations were proven to be intrinsic to the theory of operator ideals in [6, 7] and arithmetic mean operations at infinity (am-∞) were studied in [10]. Here we focus on the commutation relations between these operations and soft operations. In the process we characterize the am-interior and the am-∞ interior of an ideal. Both authors were partially supported by grants of the Charles Phelps Taft Research Center; the second named author was partially supported by NSF Grants DMS 95-03062 and DMS 97-06911.     

Primitive Ideals in the Algebra of Regular Functions on Quantum m × n Matrices

Mathematical Notes -     

Description of Closed Maximal Ideals in Topological Algebras of Continuous Vector-Valued Functions

Let X be a completely regular Hausdorff space, A be a unital locally convex algebra with jointly continuous multiplication and C(X,A) be the algebra of all continuous A-valued functions on X equipped with the topology of \({\mathcal{K}(X)}\) -convergence. Moreover, let \({\mathfrak{M}_{\ell}(A)}\) and \({\mathfrak{M}(A)}\) denote the set of all closed maximal left and two-sided ideals in A, respectively. In this note, we describe all closed maximal left and two-sided ideals in C(X,A) and show that there exist bijections from \({\mathfrak{M}_{\ell}(C(X, A))}\) onto \({X \times \mathfrak{M}_{\ell}(A)}\) and \({\mathfrak{M}(C(X, A))}\) onto \({X \times \mathfrak{M}(A)}\) . We also present new characterizations of closed maximal ideals in C(X, A) when A is a unital commutative locally convex Gelfand–Mazur algebra with jointly continuous multiplication.     

幂等剩余格

研究一种特殊的刺余格--幂等刺余格,证明满足幂等性的一般刺余格必是可换剩余格,即不存在非可换的幂等剩余格.讨论幂等剩余格的基本性质以及与各类特殊剩余格之间的关系.在一般剩余格中引入psG-滤子的概念,给出其一组等价条件,并借助psG-滤子刻画幂等刺余格的特征.     

Primary Ideals and Valuation Ideals in Semigroups

Let S be a cancellation torsion-free additive semigroup with identity 0 and let S {0}. We consider the relation between the valuation ideals and primary ideals of S. We give a characterization that each primary ideal is a valuation ideal in S and also give a characterization that each valuation ideal is a primary ideal in S.AMS Subject Classification (1991): 20M14     

Idempotent bounded C-semigroups

A semigroup with zero isidempotent bounded (IB) if it is the 0-direct union of idempotent generated principal left ideals and the 0-direct union of idempotent generated principal right ideals. Notable examples are completely 0-simple semigroups and the wider class of primitive abundant semigroups. Significant to the structure of these semigroups is that they are all categorical at zero. In this paper we describe IB semigroups that are categorical at zero in terms ofdouble blocked Rees matrix semigroups. This generalises Fountain's characterisation of primitive abundant semigroups via blocked Rees matrix semigroups [1], which in turn yields the Rees theorem for completely 0-simple semigroups.