Auslander-Regular and Cohen–Macaulay Quantum Groups

The quantized enveloping C(q)-algebra U _q (C) associated to a Cartan matirx C is Auslander-regular and Cohen–Macaulay. This is deduced from a general theorem, which also applies to solvable polynomial algebras. The results are obtained by constructing a new filtration keeping the properties of the associated graded algebra of a given multi-filtered algebra.     

A generalized Macaulay theorem and generalized face rings

We prove that the f-vector of members in a certain class of meet semi-lattices satisfies Macaulay inequalities 0?k^∂(fk)?fk−1 for all k?0. We construct a large family of meet semi-lattices belonging to this class, which includes all posets of multicomplexes, as well as meet semi-lattices with the “diamond property,” discussed by Wegner [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828], as special cases. Specializing the proof to the later family, one obtains the Kruskal-Katona inequalities and their proof as in [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828].For geometric meet semi-lattices we construct an analogue of the exterior face ring, generalizing the classic construction for simplicial complexes. For a more general class, which also includes multicomplexes, we construct an analogue of the Stanley-Reisner ring. These two constructions provide algebraic counterparts (and thus also algebraic proofs) of Kruskal-Katona's and Macaulay's inequalities for these classes, respectively.     

A Local-Global Principle for Macaulay Posets

We consider the shadow minimization problem (SMP) for Cartesian powers P ⁿ of a Macaulay poset P. Our main result is a local-global principle with respect to the lexicographic order L ⁿ . Namely, we show that under certain conditions the shadow of any initial segment of the order L ⁿ for n 3 is minimal iff it is so for n = 2. These conditions include such poset properties as additivity, shadow increasing, final shadow increasing and being rank-greedy. We also show that these conditions are essentially necessary for the lexicographic order to provide nestedness in the SMP.     

A Class of Quasipolar Rings

An element a of a ring R is called J-quasipolar if there exists p ² = p ∈ R satisfying p ∈ comm²(a) and a + p ∈ J(R); R is called J-quasipolar in case each of its elements is J-quasipolar. The class of this sort of rings lies properly between the class of uniquely clean rings and the class of quasipolar rings. In particular, every J-quasipolar element in a ring is quasipolar. It is shown, in this paper, that a ring R is J-quasipolar iff R/J(R) is boolean and R is quasipolar. For a local ring R, we prove that every n × n upper triangular matrix ring over R is J-quasipolar iff R is uniquely bleached and R/J(R) ? ?₂. Moreover, it is proved that any matrix ring of size greater than 1 is never J-quasipolar. Consequently, we determine when a 2 × 2 matrix over a commutative local ring is J-quasipolar. A criterion in terms of solvability of the characteristic equation is obtained for such a matrix to be J-quasipolar.     

Additive Macaulay Posets

An additive sequence of integers is a finite sequence in which the sum of any number of consecutive terms is less than or equal to the sum of the same number of initial terms in the sequence and greater than or equal to the sum of the same number of final terms in the sequence. If the final several terms in an additive sequence are greater than or equal to, in order, the initial several terms of a second additive sequence, then the juxtaposition of the two sequences is also additive. This simple fact has combinatorial corollaries.     

Some Characterizations of Dedekind Rings

In this article we establish some characterizations for Dedekind rings.     

Generalized Stable Regular Rings

Abstract

In this paper we show that a regular ring R is a generalized stable ring if and only if for every x ∈ R, there exist a w ∈ K(R) and a group G in R such that wx ∈ G. Also we show that if R is a generalized stable regular ring, then for any A ∈ M _n (R), there exist right invertible matrices U ₁, U ₂ ∈ M _n (R) and left invertible matrices V ₁, V ₂ ∈ M _n (R) such that U ₁ V ₁ AU ₂ V ₂ = diag(e ₁,…, e _n ) for some idempotents e ₁,…, e _n  ∈ R.     

A Generalization of Divided Domains and Its Connection to Weak Baer Going-Down Rings

Many results on going-down domains and divided domains are generalized to the context of rings with von Neumann regular total quotient rings. A (commutative unital) ring R is called regular divided if each P ∈ Spec(R)?(Max(R) ∩ Min(R)) is comparable with each principal regular ideal of R. Among rings having von Neumann regular total quotient rings, the regular divided rings are the pullbacks K× _K/P D where K is von Neumann regular, P ∈ Spec(K) and D is a divided domain. Any regular divided ring (for instance, regular comparable ring) with a von Neumann regular total quotient ring is a weak Baer going-down ring. If R is a weak Baer going-down ring and T is an extension ring with a von Neumann regular total quotient ring such that no regular element of R becomes a zero-divisor in T, then R ? T satisfies going-down. If R is a weak Baer ring and P ∈ Spec(R), then R + PR _(P) is a going-down ring if and only if R/P and R _P are going-down rings. The weak Baer going-down rings R such that Spec(R)?Min(R) has a unique maximal element are characterized in terms of the existence of suitable regular divided overrings.     

Abel正则球与完全正则半群

本文分别讨论了关于结合环和半群的二个定理，并且由结合环的这二个定理推出了如下准则：结合环Ｒ是Ａｂｅｌ正则的，当且仅当Ｒ的每个拟理想是正则环．     

Containing Symbolic Powers in Regular Rings

We extend some results of Ein–Lazarsfeld-Smith, Hochster–Huneke, and Takagi-Yoshida concerning the containments of symbolic powers of ideals in ordinary powers in regular rings.     

关于正则环的理想

本文把对于正则环的可比较性推广到对于正则环上理想的可比较性,得到这 种可比较性的若干刻画.进而我们证明,对稳定秩1的理想, Roth问题有肯定解答.     

结合环类中的E-单环

在全部结合环类中，对幂等根Ｅ，定义了Ｅ－单环．本文给出Ｅ－单环的一个刻画．     

一类有限环

本文指出了文 [5 ]的一些错误 ,给出了文 [5 ]定理 1的一个反例 .并给出了一类交换环的等价条件 .     

矩阵环中的一类极大的广义的Armendariz子环

An associative ring with identity R is called Armendariz if, whenever （∑^m i=0^aix^i）（∑^n j=0^bjx^j）=0 in R[x],aibj=0 for all i and j. An associative ring with identity is called reduced if it has no non-zero nilpotent elements. In this paper, we define a general reduced ring （with or without identity） and a general Armendariz ring （with or without identity）, and identify a class of maximal general Armendariz subrings of matrix rings over general reduced rings.     

一类环上HX环的结构

自李洪兴１９９１年提出了ＨＸ环以来，人们一直有这么一个问题没解决，就是是否存在非平凡的ＨＸ环的例子？但至今既没找到非平凡的ＨＸ环，也没有证明任一环Ｒ仅存在平凡的ＨＸ环。针对这个问题，本文提出并证明了一类环仅有平凡ＨＸ环，还给出了一系列的结构定理。这样，既为证明任一环Ｒ仅有平凡的ＨＸ环的猜想有新的启示，也为人们指明无须在这一类环上寻找非平凡ＨＸ环。     

On the Finiteness of Noetherian Rings with Finitely Many Regular Elements

Let R be a left Noetherian ring and ZD(R) be the set of all zero-divisors of R. In this paper, it is shown that if R \ ZD(R) is finite, then R is finite.     

Separativity of Regular Rings in Which 2 Is Invertible

A regular ring R is separative provided that for all finitely generated projective right R-modules A and B, A⊕ A? A⊕ B? A⊕ B implies that A? B. We prove, in this article, that a regular ring R in which 2 is invertible is separative if and only if each a ∈ R satisfying R(1 ? a ²)R = Rr(a) = ?(a)R and i(End _R (aR)) = ∞ is unit-regular if and only if each a ∈ R satisfying R(1 ? a ²)R ∩ RaR = Rr(a) ∩ ?(a)R ∩ RaR and i(End _R (aR)) = ∞ is unit-regular. Further equivalent characterizations of such regular rings are also obtained.     

Separative Cancellation over Regular Rings

This article develops element-wise characterizations of separative ideals in regular rings. These give connections among separativity, annihilators, cokernels and unit-regularity. For strongly separativity, the analogs are also obtained.     

Macaulay durations for nonparallel shifts

Macaulay duration is a well-known and widely used interest rate risk measure. It is commonly believed that it only works for parallel shifts of interest rates. We show in this paper that this limitation is largely due to the traditional parametric modelling and the derivative approach, the Macaulay duration works for non-parallel shifts as well when the non-parametric modelling and the equivalent zero coupon bond approach are used. We show that the Macaulay duration provides the best one-number sensitivity information for non-parallel interest rate changes and that a Macaulay duration matched portfolio is least vulnerable to the downside risk caused by non-parallel rate changes under some verifiable conditions. AMS Classification 65K10 · 90C90