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     

BCH-代数的拟结合理想和拟结合模糊理想

在BCH-代数中,引入了拟结合理想和模糊拟结合理想的概念.研究了BCH-代数中模糊拟结合理想的相关性质,给出了BCH-代数拟结合理想的构造.     

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.     

Fuzzy Ideals and Fuzzy Interior Ideals in Ordered Semirings

The notions of fuzzy ideals and fuzzy interior ideals in ordered semirings are introduced and some of their characterizations are obtained through regularity criterion and with some operations on them.     

环中的粗素理想与模糊粗素理想

首次提出了环中的粗素理想与模糊粗素理想的概念，并讨论了它的性质及同态问题。     

半群中的粗糙模糊理想及粗糙模糊素理想

将粗糙集的理论方法用于模糊半群的研究。给出上(下)粗模糊子半群及上(下)粗模糊左(右)理想、粗糙模糊素理想等概念,研究它们的有关性质。证明上粗模糊子半群与上粗模糊理想是通常的模糊子半群与模糊理想概念的扩张。     

Ideals in Graph Algebras

We show that the graph construction used to prove that a gauge-invariant ideal of a graph C ^???-algebra is isomorphic to a graph C ^???-algebra, and also used to prove that a graded ideal of a Leavitt path algebra is isomorphic to a Leavitt path algebra, is incorrect as stated in the literature. We give a new graph construction to remedy this problem, and prove that it can be used to realize a gauge-invariant ideal (respectively, a graded ideal) as a graph C ^???-algebra (respectively, a Leavitt path algebra).     

Ideals in distributive posets

We prove that any ideal in a distributive (relative to a certain completion) poset is an intersection of prime ideals. Besides that, we give a characterization of n-normal meet semilattices with zero, thus generalizing a known result for lattices with zero.     

Adjacent Ideals to Simple Complete Ideals in Two-Dimensional Regular Local Rings

Let R be a two-dimensional regular local ring with infinite residue field, and ? be a simple complete residually rational ideal of R of order r which determines R _h . Let 𝒯 be the set of quadratic transforms T of R _h with [T: R _h ] = 1, and 𝒮 the set of simple complete ideals of R of order r which are adjacent to ? from below. If R _h is free respectively a satellite, then there exist T* ∈ 𝒯 respectively T*, T** ∈ 𝒯 and a bijective map between the set 𝒮 and the set 𝒯?{T*} respectively 𝒯?{T*, T**}.     

Full Ideals

Contractedness of 𝔪-primary integrally closed ideals played a central role in the development of Zariski's theory of integrally closed ideals in two-dimensional regular local rings (R, 𝔪). In such rings, the contracted 𝔪-primary ideals are known to be characterized by the property that I: 𝔪 = I: x for some x ∈ 𝔪 ?𝔪². We call the ideals with this property full ideals and compare this class of ideals with the classes of 𝔪-full ideals, basically full ideals, and contracted ideals in higher dimensional regular local rings. The 𝔪-full ideals are easily seen to be full. In this article, we find a sufficient condition for a full ideal to be 𝔪-full. We also show the equivalence of the properties full, 𝔪-full, contracted, integrally closed, and normal, for the class of parameter ideals. We then find a sufficient condition for a basically full parameter ideal to be full.     

L-预余拓扑空间中的理想

说明L-预余拓扑空间是L-余拓扑空间的推广;定义L-预余拓扑空间中理想间的强弱关系并给出一些相关结论;讨论L-预余拓扑空间中极大理想的性质.     

半群中的粗模糊理想

将粗糙集的理论方法用于模糊半群的研究。首先,给出上(下)粗模糊子半群及上(下)粗模糊理想等概念,并研究了它们的有关性质。随后,证明了上粗模糊子半群与上粗模糊理想是通常的模糊子半群与模糊理想概念的扩张。特别地,运用粗糙集的思想方法研究半群上的模糊同余及其截关系,给出它们的一些基本性质。     

极大本质左理想是理想的SF－环

由Ｒａｍａｍｕｒｔｈｉ和Ｍｉｎｇ的两个公开问题所推动，本文证明了如下结果：（１）如果Ｒ是ＭＥＬＴ，ＳＦ－环，那么Ｒ是正则环；（２）如果Ｒ是ＭＥＬＴ，左ＣＥ－内射，右ＳＦ－环，那么Ｒ是具有有界指数的左和右自内射正则，左和右Ｖ－环．这就给出了Ｒａｍａｍｕｒｔｈｉ和Ｍｉｎｇ两个公开问题的部分回答．     

Constructible Ideals

Based on the study of simplicial complexes, one may naturally define the constructible monomial ideals. We connect the square-free constructible ideal with the Stanley–Reisner ideal of the Alexander dual associated to a constructible simplicial complex. We give some properties of constructible ideals, and we compute the Betti numbers. We prove that all monomial ideals with linear quotients are constructible ideals. We also show that all constructible ideals have a linear resolution.     

亚BCI-代数的模糊理想

引入了亚BCI-代数的模糊子代数、模糊理想、闭模糊理想和模糊P-理想的概念,研究了它们的性质。证明了模糊子代数(模糊理想、闭模糊理想、模糊P-理想)的同态像与同态原像仍能成为模糊子代数(模糊理想、闭模糊理想、模糊P-理想)。     

Quantale的理想及由理想生成的同余

给出了Quantale的理想的概念,讨论了Quantale的由任意子集生成的理想的具体结构, 讨论了由理想生成的同余,得到了Quantale的理想的若干性质.     

Ideal Submodules Versus Ternary Ideals Versus Linking Ideals

We show that ideal submodules and closed ternary ideals in Hilbert modules are the same. We use this insight as a little peg on which to hang a little note about interrelations with other notions regarding Hilbert modules. In Section 3, we show that the ternary ideals (and equivalent notions) merit fully, in terms of homomorphisms and quotients, to be called ideals of (not necessarily full) Hilbert modules. The properties to be checked are intrinsically formulated for the modules (without any reference to the algebra over which they are modules) in terms of their ternary structure. The proofs, instead, are motivated from a third equivalent notion, linking ideals (Section 2), and a Theorem (Section 3) that all extends nicely to (reduced) linking algebras. As an application, in Section 4, we introduce ternary extensions of Hilbert modules and prove most of the basic properties (some new even for the known notion of extensions of Hilbert modules), by reducing their proof to the well-known analogue theorems about extensions of C^?–algebras. Finally, in Section 5, we propose several open problems that our method naturally suggests.

     

Prime Ideals in Near-Rings

An ideal I of a near-ring R is a type one prime ideal if whenever a Rb ? I, then a ∈ I or b ∈ I. This paper considers the interconnections between prime ideals and type one prime ideals in near-rings. It also develops properties of type one prime ideals, gives several examples illustrating where prime and type one prime are not equivalent, and investigates the properties of the type one prime radical. Several different types of conditions are given which guarantee that a prime ideal is type one. The class of all near-rings for which each prime ideal is type one is investigated and many examples of such near-rings are exhibited. Various localized distributivity conditions are found which are useful in establishing when prime ideals will be type one prime.