代数整数环的商环

In this paper,we discuss on the algebraic interger ring of the quadratic field of Q.The form of the quotient ring of the algebraic interger ring with respect to the order ideal is presented.     

关于环Z[(√c)]的商环的注记

讨论了一类环的商环元素的个数,并推广了文[1,2]的结果.     

关于环Z[√c]的商环的注记

讨论了一类环的商环元素的个数，并推广了交[1,2]的结果。     

关于环Z[m~（1/2）]的商环元素个数的一个新的证明

文献[1]热运用环论的方法证明了环Z[m~（1/2）]热的商环Z[m~（1/2）]/（a＋bm~（1/2））的元素个数是｜a2-b2m｜.我们将用主理想整环上的模的理论给出一种简洁的证明.     

关于高斯的一点注记

评价科学上的历史人物,学术界往往见仁见智,各有所执,这是常有的事。前些时读了陈庆益教授的“论高斯”一文(载本刊1984年第4期,145—147页),觉其所作论评不同凡响。诚然,正如该文所述,其观点与西方M.Kline是一致的,即认为Gauss主要是一位面向过去,更甚于面向未来的历史过渡性人物。Gauss的突出贡献是在数沦方面,而数论并不是19世纪的数学主流。当时的主流应是分析数学(变量数学)。无疑,对分析学主流的提法是符合历史事实的。但是本文乐于指出,该文忍略了Gauss在分析学领域的一     

高斯整数环上的二阶线性群

<正> 记Z[i]为高斯整数环,G_2=GL(2,Z[i]),G_2~±={X∈G_2|det X=±1},G_2~±=SL(2,Z[i]),G_2和G_2~±的投影群记为PG_2和PG_2~±.(注意G_2~±的投影群等于PG_2~±,这很容易证明.)任一X∈G_2在PG_2中的像记为εX,任一X∈G_2~±在PG_2~±中的像记为±X,任一群G的自同构群记为A(G),G的换位子群记为G’.在全文中记X为X中元素取复数共轭所得之阵,并记     

关于超素环的左经典商环

In this paper,we define the left rank of a ring R and discuss the relatlons between a left ore reng R and its left classical quotient ring Q on the properties of their ideals and left ranks.Also we gave four equivalent types of the proposition:a ring R has a left classical quotient ring Q and Q is a division ring.As an application of this result,we gave a brief proof method which different from [2] on the theorem of the Goldie Prime rings.     

关于Buchsbaum环的注记

本文证明了下述结论，设Ａ是一个级数为ｄ的Ｂｕｃｈｓｂａｕｍ环，（ａ₁，ａ₂，…，ａ_n）是Ａ的一个参数系统，则任何正整数ｎ，Ａ／（ａ₁，ａ₂，…，ａ_k）ⁿ（１≤ｋ≤ｄ）仍是ｄ－ｋ维的Ｂｕｃｈｓｂａｕｍ环．     

一个关于高斯整数的最小正整数问题

主要讨论了高斯整数多项式所表整数的计算,利用初等数论基本理论和方法,获得了一个含有n(n≥1)个高斯整数常量和变量的线性多项式虚部所表整数中最小正整数的精确显式表达,并进一步获得了该多项式虚部所表整数的全体.     

The Quotient Ring of Algebraic Interger Ring

ＴｈｅＱｕｏｔｉｅｎｔＲｉｎｇｏｆＡｌｇｅｂｒａｉｃＩｎｔｅｒｇｅｒＲｉｎｇ￥ＤｉｎｇＲｕｉｘｉａｎｇ；ＬｉｕＧｕａｎｇｌｉａｎｇ（ＰｕｙａｎｇＥｄｕｃａｔｉｏｎｓＣｏｌｌｅｇｅ，Ｈｅｎａｎ，４５７０００）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｅｐａｐｅｒ，ｗｅ...     

Conjectures on the Quotient Ring by Diagonal Invariants

We formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring in two sets of variables by the ideal generated by all S _n invariant polynomials without constant term. The theory of the corresponding ring in a single set of variables X = {x ₁, ..., x _n} is classical. Introducing the second set of variables leads to a ring about which little is yet understood, but for which there is strong evidence of deep connections with many fundamental results of enumerative combinatorics, as well as with algebraic geometry and Lie theory.     

The Largest Left Quotient Ring of a Ring

The left quotient ring (i.e., the left classical ring of fractions) Q_cl(R) of a ring R does not always exist and still, in general, there is no good understanding of the reason why this happens. In this article, existence of the largest left quotient ring Q_l(R) of an arbitrary ring R is proved, i.e., Q_l(R) = S₀(R)^?1R where S₀(R) is the largest left regular denominator set of R. It is proved that Q_l(Q_l(R)) = Q_l(R); the ring Q_l(R) is semisimple iff Q_cl(R) exists and is semisimple; moreover, if the ring Q_l(R) is left Artinian, then Q_cl(R) exists and Q_l(R) = Q_cl(R). The group of units Q_l(R)* of Q_l(R) is equal to the set {s^?1t | s, t ∈ S₀(R)} and S₀(R) = R ∩ Q_l(R)*. If there exists a finitely generated flat left R-module which is not projective, then Q_l(R) is not a semisimple ring. We extend slightly Ore's method of localization to localizable left Ore sets, give a criterion of when a left Ore set is localizable, and prove that all left and right Ore sets of an arbitrary ring are localizable (not just denominator sets as in Ore's method of localization). Applications are given for certain classes of rings (semiprime Goldie rings, Noetherian commutative rings, the algebra of polynomial integro-differential operators).     

A Note on the Number of Distinct Distances

We refine a method introduced in [1] and [2] for studying the number of distinct values taken by certain polynomials of two real variables on Cartesian products. We apply it to prove a "gap theorem", improving a recent lower bound on the number of distinct distances between two collinear point sets in the Euclidean space. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Note on the Domination Number of Triangulations

In this note, we prove that every triangulation G on any closed surface has domination number at most . This unifies some results on the domination number of a triangulation on a closed surface.     

关于环同态扩张的注记

本文指出环同态扩张的几个新的充分条件．     

关于Volterra积分方程的一个注记

我们减弱了Mydlarczyk W的定理条件，得到他的相同的结果。     

A Note on a Marcinkiewicz Integral Operator

We prove the L_p boundedness of Marcinkiewicz integral operators with rough kernels on ℝⁿ and T ⁿ.