Ideals and atoms of <Emphasis Type="Italic">BZ</Emphasis>-algebras

Ideals and atoms are studied by various authors from different point of views. In different algebras there are studied different ideals, but obtained results are similar. Below we present a new method of study of ideals in BZ-algebras. Using this method we describe the connection between ideals of various types. Supported by Natural Science Foundation of China (Grant No. 60775038).     

格的TL-模糊理想

本文研究了格的TL-模糊理想. 利用生成TL-模糊理想, 证明了一个模格的全体T_M-模糊理想形成一个完备的模格. 此外, 利用L-模糊集的投影和截影, 获得了将直积格的TL-模糊理想表示成分量格的TL-模糊理想的L-直积的一个充分必要条件. 所得结果进一步推广和发展了格的模糊理想的理论.     

Relative polars in ordered sets

In the paper, the notion of relative polarity in ordered sets is introduced and the lattices of R-polars are studied. Connections between R-polars and prime ideals, especially in distributive sets, are found.     

On L-fuzzy ideals in semirings II

We study some properties of L-fuzzy left (right) ideals of a semiring R related to level left (right) ideals     

Serre's Condition Rℓ for Affine Semigroup Rings

In this note we characterize the affine semigroup rings K[S] over an arbitrary field K that satisfy condition R_? of Serre. Our characterization is in terms of the face lattice of the positive cone pos(S) of S. We start by reviewing some basic facts about the faces of pos(S) and consequences for the monomial primes of K[S]. After proving our characterization we turn our attention to the Rees algebras of a special class of monomial ideals in a polynomial ring over a field. In this special case, some of the characterizing criteria are always satisfied. We give examples of non-normal affine semigroup rings that satisfy R₂.     

Reliability Approximation for Markov Chain Imbeddable Systems

In the present article, a simple method is developed for approximating the reliability of Markov chain imbeddable systems. The approximating formula reduces the problem to the reliability assessment of smaller systems with structure similar to the original systems. Two specific reliability structures which have attracted considerable research interest recently (r-within-consecutive-k-out-of-n system and two dimensional r-within-k ₁ × k ₂-out-of-n ₁ × n ₂ system) are studied by the new approach and numerical calculations are carried out, which reveal the high quality of our approximations. Several possible extensions and generalizations are also presented in brief.     

Ideal Structure of Leavitt Path Algebras with Coefficients in a Unital Commutative Ring

For a (countable) graph E and a unital commutative ring R, we analyze the ideal structure of the Leavitt path algebra L _R (E) introduced by Mark Tomforde. We first modify the definition of basic ideals and then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of L _R (E), and we then determine prime graded basic ideals and left (or right) primitive graded ideals of L _R (E). In particular, when E satisfies Condition (K) and R is a field, they imply that the set of prime ideals and the set of primitive ideals of L _R (E) coincide.     

F_2+vF_2上的循环码

本文研究了环F_2+vF_2上的循环码.利用标准形生成元集刻画了环F_2+vF_2上的循环码的代数结构,证明了环F_2+vF_2上的每一个非零的循环码均有唯一的标准形生成元集,进而得到了每一个循环码均是由一个多项式生成的.     

Connections between stability and asymptotic stability of nonlinear switched systems

Here are considered nonlinear switched systems in which the switching occurs among a class of subsystems that are characterized by input–output properties stated in terms of L_p spaces of signals. The relationships between the L_p stability of each subsystem and the internal stability of the switched system are studied. In particular, conditions on the dwell time of the switching signals that guarantee the asymptotic stability of the overall system are provided. The connections among these conditions and the L_p input–output properties of the subsystems are investigated.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-mixed intersection bodies

In this paper the author first introduce a new concept of L _p -dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of L _p intersection body to L _p -mixed intersection body. Inequalities for L _p -dual mixed volumes of L _p -mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities. This work was supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. Y605065) and the Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

On n-Absorbing Ideals of Commutative Rings

Let R be a commutative ring with 1 ≠ 0 and n a positive integer. In this article, we study two generalizations of a prime ideal. A proper ideal I of R is called an n-absorbing (resp., strongly n-absorbing) ideal if whenever x ₁…x _n+1 ∈ I for x ₁,…, x _n+1 ∈ R (resp., I ₁…I _n+1 ? I for ideals I ₁,…, I _n+1 of R), then there are n of the x _i 's (resp., n of the I _i 's) whose product is in I. We investigate n-absorbing and strongly n-absorbing ideals, and we conjecture that these two concepts are equivalent. In particular, we study the stability of n-absorbing ideals with respect to various ring-theoretic constructions and study n-absorbing ideals in several classes of commutative rings. For example, in a Noetherian ring every proper ideal is an n-absorbing ideal for some positive integer n, and in a Prüfer domain, an ideal is an n-absorbing ideal for some positive integer n if and only if it is a product of prime ideals.     

Idempotents in n-ary Semigroups

We select a new type of elements of n-ary (n 3) semigroups with an idempotent and investigate properties of ideals connected with these elements.AMS Subject Classification (2000) 20N15     

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.     

Relative MilnorK-theory

LetR be a commutative, semi-local ring,I ₁, ...,I _s ideals. In this paper, we define therelative Milnor K-groups of (R;I ₁, ...,I _s ),K _p ^M (R;I ₁, ...,I _s ), and show that these groups have many of the properties of the usual MilnorK-groups of a field. In particular, assuming a weak condition on the ideals, we show thatK _p ^M (R;I ₁, ...,I _s ) is isomorphic to the weightp portion of the relative QuillenK-groupK _p (R;I ₁, ...,I _s ), after inverting (p–1)!. We also define the relative group homology of GL _n (R;I ₁, ...,I _s ), and show thatK _p ^M (R;I ₁, ...,I _s ) is isomorphic toH _p (GL_p(R;I ₁, ...,I _s ))/Im(H _p (GL _p–1 (R;I ₁, ...,I _s ))). Finally, we consider a generalization to the relative setting of Kato's conjecture asserting that the Galois symbol gives an isomorphism fromK _p ^M (F)/l ^v to , and show that this relative version of Kato's conjecture implies the Quillen-Lichtenbaum conjectures asserting the Chern class:

Generalizations of Prime Ideals

Let R be a commutative ring with identity. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R is weakly prime (resp., almost prime) if a, b ∈ R with ab ∈ I ? {0} (resp., ab ∈ I ? I ²) implies a ∈ I or b ∈ I. Let φ:?(R) → ?(R) ∪ {?} be a function where ?(R) is the set of ideals of R. We call a proper ideal I of R a φ-prime ideal if a, b ∈ R with ab ∈ I ? φ(I) implies a ∈ I or b ∈ I. So taking φ_?(J) = ? (resp., φ₀(J) = 0, φ₂(J) = J ²), a φ_?-prime ideal (resp., φ₀-prime ideal, φ₂-prime ideal) is a prime ideal (resp., weakly prime ideal, almost prime ideal). We show that φ-prime ideals enjoy analogs of many of the properties of prime ideals.     

Monoids and Groups of <Emphasis Type="Italic">I</Emphasis>-Type

A monoid S generated by {x₁,. . .,x_n} is said to be of (left) I-type if there exists a map v from the free Abelian monoid FaM_n of rank n generated by {u₁,. . .,u_n} to S so that for all a∈FaM_n one has {v(u₁a),. . .,v(u_na)}={x₁v(a),. . .,x_nv(a)}. Then S has a group of fractions, which is called a group of (left) I-type. These monoids first appeared in the work of Gateva-Ivanova and Van den Bergh, inspired by earlier work of Tate and Van den Bergh. In this paper we show that monoids and groups of left I-type can be characterized as natural submonoids and groups of semidirect products of the free Abelian group Fa_n and the symmetric group of degree n. It follows that these notions are left–right symmetric. As a consequence we determine many aspects of the algebraic structure of such monoids and groups. In particular, they can often be decomposed as products of monoids and groups of the same type but on less generators and many such groups are poly-infinite cyclic. We also prove that the minimal prime ideals of a monoid S of I-type, and of the corresponding monoid algebra, are principal and generated by a normal element. Further, via left–right divisibility, we show that all semiprime ideals of S can be described. The latter yields an ideal chain of S with factors that are semigroups of matrix type over cancellative semigroups. In memory of Paul Wauters Mathematics Subject Classifications (2000) 20F05, 20M05; 16S34, 16S36, 20F16. The authors were supported in part by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium), Flemish–Polish bilateral agreement BIL 01/31, and KBN research grant 2P03A 033 25 (Poland).     

Localization of <Emphasis Type="Italic">LM</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-algebras

The aim of this paper is to define the localization LM _n -algebra of an LM _n —algebra L with respect to a topology F on L; in Section 5 we prove that the maximal LM _n -algebra of fractions (defined in [3]) and the LM _n -algebra of fractions relative to an Λ—closed system (defined in Section 2) are LM _n -algebras of localization.     

Generalized fuzzy ideals of near-rings

The concept of（∈,∈∨q）-fuzzy subnear-rings（ideals） of a near-ring is introduced and some of its related properties are investigated.In particular,the relationships among ordinary fuzzy subnear-rings（ideals）,（∈,∈∨ q）-fuzzy subnear-rings（ideals） and（∈,∈∨q）-fuzzy subnear-rings（ideals） of near-rings are described.Finally,some characterization of [μ]t is given by means of（∈,∈∨ q）-fuzzy ideals.     

The L p -curvature images of convex bodies and L p -projection bodies

Associated with the L _p -curvature image defined by Lutwak, some inequalities for extended mixed p-affine surface areas of convex bodies and the support functions of L _p -projection bodies are established. As a natural extension of a result due to Lutwak, an L _p -type affine isoperimetric inequality, whose special cases are L _p -Busemann-Petty centroid inequality and L _p -affine projection inequality, respectively, is established. Some L _p -mixed volume inequalities involving L _p -projection bodies are also established.     

R 0-algebras and weak dually residuated lattice ordered semigroups

We introduce the notion of weak dually residuated lattice ordered semi-groups (WDRL-semigroups) and investigate the relation between R ₀-algebras and WDRL-semigroups. We prove that the category of R ₀-algebras is equivalent to the category of some bounded WDRL-semigroups. Moreover, the connection between WDRL-semigroups and DRL-semigroups is studied.