UNIQUE FACTORIZATION AND BIRTH OF ALMOST PRIMES

ABSTRACT

In studying unique factorization of domains we encountered a property of ideals. Using that we define the notion of almost prime ideals and prove that in Noetherian domains almost prime ideals are primary. We also prove that in a regular domain almost primes are precisely primes. Further, we define strictly nonprime ideals and study some inter relations between almost prime ideals, strictly nonprime ideals and factorization of ideals.     

NOETHERIAN GT-REGULAR RINGS ARE REGULAR

ＮＯＥＴＨＥＲＩＡＮＧＴ－ＲＥＧＵＬＡＲＲＩＮＧＳＡＲＥＲＥＧＵＬＡＲ￥ＬＩＨＵＩＳＨＩ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈａａｎｘｉＮｏｒｍａｌＵｎｉｖｉｓｉｔｙＸｉ＇ａｎ７１００６２，Ｃｈｉｎａ．）Ａｂｓｔｒａｃｔ：Ｉｔｉｓｐｒ...     

LIGHT FACTORIZATION STRUCTURES

Abstract

In this paper we investigate, for connection subcategories A of a topological category K, the concepts of A-monotone quotients and A-light sources, and characterize (1) those A, which give rise to an (A-monotone quotient, A-light)- factorization structure on K, (2) those factorization structures (C,D) on K, which are light, i.e. of the form (A-monotone quotient, A-light) for suitable A. It turns out that light factorization structures are rather rare in Top, but abundant and well-behaved in categories with hereditary quotients.     

关于完全环与Noether环

（ｉ）环Ｒ是左完全环，当且仅当存在一个基数ｃ，使得任意平坦左Ｒ－模是一个拟投射模和一个ｃ－限制的ＥＳ－模的直和。（ｉｉ）Ｒ是左Ｎｏｅｔｈｅｒ环，当且仅当存在一个基数ｃ，使得任意内射左Ｒ－模的直和是一个（拟）连续模和一个ｃ－限制的ＥＳ－模的直和。     

LEFT-SYMMETRIC ALGEBRAS AND COMPLETE LIE ALGEBRAS

ABSTRACT

In this paper, we discuss compatible left-symmetric algebra structures on some complete Lie algebras, and as an application, we obtain all the derivations of such left-symmetric algebras.     

FACTORIZATION IN MONOID DOMAINS

In this paper, we determine necessary and sufficient conditions for the group ring D[G] to be a BFD (resp., an FFD, an SFFD). Also we give necessary and sufficient conditions for the monoid domain D[S] to be a BFD (resp., an FFD, an SFFD). In addition, we characterize when themonoid domain D[S] is a UFD in terms of 2-factoriality.     

交换半群环的分式环

设含幺交换环Ｒ对其乘法子集Ｔ的分式环为ＲＴ，交换幺半群Ｓ在其子半群∑处局部化为Ｓ∑本文证明了Ｒ［Ｓ］对于Ａ的分环式环Ｒ［Ｓ］ＡＭ 构于半群环ＲＴ［Ｓ∑］。     

FACTORIZATION PROPERTIES OF MODULE HOMOMORPHISMS

Abstract

The aim of this note is to apply various factorization properties of module homomorphisms to establish criteria for projectivity and quasi-projectivity.     

一类模李代数的导子代数(英文)

本文通过构造一类模线状李代数,求出了它的导子代数,并且证明这个导子代数是可解但不完备的模李代数.这将有利于研究一般模线状李代数的结构.     

LAX OPERATOR ALGEBRAS AND GRADINGS ON SEMI-SIMPLE LIE ALGEBRAS

A Lax operator algebra is constructed for an arbitrary semi-simple Lie algebra over ? equipped with a ?-grading, and arbitrary compact Riemann surface with marked points. In this set-up, a treatment of almost graded structures, and classification of the central extensions of Lax operator algebras are given. A relation to the earlier approach based on the Tyurin parameters is established.     

SEMIGROUP GRADINGS OF FULL MATRIX RINGS

Let F be a field, and let A = M_n (F) be the algebra of all n × n matrices over F. For each finite semigroup S, we describe all gradings A = ⊕ _s∈S A _S of A by S such that all standard matrix units e_ij of A are homogeneous elements of A.     

THE ADJOINT SEMIGROUP OF A RING

Abstract

We classify cross-sections of the ? and ? Green's relations on the finite symmetric inverse semigroup ?𝒮 _n , determine which of them are isomorphic, and study their disposition with respect to the ?-cross-sections of ?𝒮 _n .     

偏代数族的积偏代数

定义了一族偏代数物积偏代数的概念，且对偏代数间的同态映射，由同态映射诱导的合同关系及由同态映射和合同关系诱导的合同关系与积偏代数一起作了深入的讨论，分别得到了一些有意义的结果。     

可换环上辛代数与一般线性李代数之间的中间李代数

本文研究了含幺可换环上一般线性李代数的子代数结构.通过构造特殊矩阵并利用这些矩阵进行计算, 得到了任意含幺可换环上辛代数与一般线性李代数之间的所有中间李代数的形式.并且有利于研究可换环上相应的典型群的子群结构.     

DISTINGUISHED PRE-NICHOLS ALGEBRAS

We formally define and study the distinguished pre-Nichols algebra \( \tilde{B} \)(V) of a braided vector space of diagonal type V with finite-dimensional Nichols algebra B(V). The algebra \( \tilde{B} \)(V) is presented by fewer relations than B(V), so it is intermediate between the tensor algebra T(V) and B(V). Prominent examples of distinguished pre-Nichols algebras are the positive parts of quantized enveloping (super)algebras and their multiparametric versions. We prove that these algebras give rise to new examples of Noetherian pointed Hopf algebras of finite Gelfand-Kirillov dimension. We investigate the kernel (in the sense of Hopf algebras) of the projection from \( \tilde{B} \)(V) to B(V), generalizing results of De Concini and Procesi on quantum groups at roots of unity.