分式化时模理论的一些稳定性性质的保持性

本文讨论了将分式环S－1R上的模归约到R上的模时包（hull）、类（class）、界（bound）、秩（rank）的保持性,证明了：1）NR是A在MR中的包当且仅当NS－1R是A在MS－1R中的包;2）还讨论了分式模（S－1M）S－1R与MR之间上述性质存在单向或双向保持性的条件．     

J-semicommutative环的性质

环冗称为J—semicommutative若对任意B,b∈R由ab=0可以推得aRb∈J（R）,这里J（R）是环R的Jacobson根．环R是J—semicommutative环当且仅当它的平凡扩张是J—semicommutative环当且仅当它的Don＇oh扩张是J—semicommutative环当且仅当它的Nagata扩张是,一semicommutative环当且仅当它的幂级数环是J—semicommutative环．若R／J（R）是semicommutative环,则可得到R是J-semicommutative环．本文进一步论证了如果,是环月的一个幂零理想,且R／I是J—semicommutative环,则R也是J-semicommutative环最后给出了J—semicommutative环与其他一些常见环的联系     

关于左A—内射环

本文主要证明了：（1）适合右零化子升链条件的左A－内射环为QF环。（2）适合左零化子升链条件的左f-内射环为QF环。（3）若对环R的任意左理想A，B和右理想I满足r(A∩B）=r(A) r(B),rι(I)=I，则R为半完全环且有本质左基座，特别地，右CF的左A－内射环（或E(RR)为投射左R－模）为QF环。     

广义幂级数环的PP性质

作为幂级数环的推广,Ｒｉｂｅｎｂｏｉｍ引入了广义幂级数环的概念．设Ｒ是有单位元的交换环,（Ｊ,≤）是严格全序半群．本文中我们证明了如下结果：（１）广义幂级数环 ［［Ｒｓ］］是ＰＰ－环当且仅当Ｒ是ＰＰ－环且Ｂ（Ｒ）的任意 Ｓ－可标子集Ｃ在Ｂ（Ｒ）中有最小上界;（２）如果对任意ｓ∈Ｓ都有０≤ｓ,则［［Ｒｓ,≤］］是弱ＰＰ－环当且仅当Ｒ是弱ＰＰ－环．我们还给出了一个例子说明交换的弱ＰＰ－环可以不是ＰＰ－环．     

广义线性McCoy环

称环R是右线性McCoy的,如果R[x]中非零线性多项式f（x）,g（x）满足I（x）g（x）=0,则存在非零元素r∈R使得f（x）r=0．设a是环R的自同态,通过用斜多项式环R[x;a]中的元素代替一般多项式环R[x]中的元素而引入a-线性McCoy环的概念．讨论了a-线性McCoy环的基本性质和扩张性质．     

投射可迁环上矩阵环的若当同态

设R′是一个环,Mn′（R′）是R′上的n′×n′矩阵环.如果环R有不变基数性质并且每个有限生成的投射左R-模是自由模,则R是一个投射自由环.如果环R≌Mr（S）,其中S是一个投射自由环,则R是一个投射可迁环.当R是一个投射可迁环时,给出了从Mn′（R′）到Mn（R）（n′≥n≥2）的若当同态的代数公式.     

RP—环

本文定义了ＲＰ－环,讨论了它的性质和等价条件,得出了Ｓ＝Ｍｎ（Ｒ）是ＲＰ－环妥且仅当Ｒ是ＲＰ－环;Ｒｉ是ＲＰ－环当且仅当每个Ｒｉ是ＲＰ－环．最后讨论了ＲＰ－环的同态象和左投射维数．     

一般环之分式环

本文给出一般环R之分式环△^-1R的构造和△^-R的理想的一些性质,推广了现有中外教科书的结论．证明了（1）△^-1R中理想有且仅有形式△^-1I,其中△3R．（2）如果P是R的素理想,且△∩P=Ф,则△^-1P是△^-1R的素理想,且θ^-1（△^-1P）=P。     

Malcev-Neumann环的主拟Baer性质

设R是环，G是偏序群，σ是从G到R的自同构群的映射。本文研究了Malcev-Neumann环R*((G))是主拟Baer环的条件。证明了如下结果：如果R是约化环并且σ是弱刚性的，则R*((G))是主拟Baer环当且仅当R是主拟Baer环，并且I(R)的任意G可标子集在I(R)中具有广义并．     

罗朗级数环的主拟Baer性

称环 R为右主拟 Baer环（简称为右p·q.Baer环）,如果 R的任意主右理想的右零化子可由幂等元生成.本文证明了,若环 R满足条件Sl（R）（?）C（R）,则罗朗级数环R[[x,x-1]]是右p.q.Baer环当且仅当R是右p.q.Baer环且R的任意可数多个幂等元在I（R）中有广义join.同时还证明了,R是右p.q.Baer环当且仅当R[x,x-1]是右P.q.Baer环.     

环的投射生成元和K0群

设Ｒ是含幺结合环,Ｐｇ（Ｒ）是Ｒ的所有投射生成元的同构类组成的半群,Ｇｒ（Ｐｇ（Ｒ））是Ｐｇ（Ｒ）的Ｇｒｏｔｈｅｎｄｉｅｃｋ群,在本文中我们证明了Ｋ０（Ｒ）＝Ｇｒ（Ｐｇ（Ｒ））。由此我们得到对任意ＶＢＮ环Ｒ,存在环Ｓ满足Ｓ＾２＝Ｓ并且具有Ａｕｔ－Ｐｉｃ性质,最后我们给出了环的一个分类,并且用Ｐｇ（Ｒ）的周期性对它作了描述。     

Nil-Clean Rings with Involution

A *-ring R is called a nil *-clean ring if every element of R is a sum of a projection and a nilpotent.Nil *-clean rings are the *-version of nil-clean rings introduced by Diesl.This paper is about the nil *-clean property of rings with emphasis on matrix rings.We show that a *-ring R is nil *-clean if and only if J(R) is nil and R/J(R) is nil*-clean.For a 2-primal *-ring R,with the induced involution given by (aij)* =(a*ij)T,the nil *-clean property of Mn(R) is completely reduced to that of Mn(Z2).Consequently,Mn(R) is not a nil *-clean ring for n =3,4,and M2(R) is a nil *-clean ring if and only if J(R) is nil,R/J(R) is a Boolean ring and a*-a ∈ J(R) for all a ∈ R.     

Ideal-Symmetric and Semiprime Rings

Lambek extended the usual commutative ideal theory to ideals in noncommutative rings, calling an ideal A of a ring R symmetric if rst ∈ A implies rts ∈ A for r, s, t ∈ R. R is usually called symmetric if 0 is a symmetric ideal. This naturally gives rise to extending the study of symmetric ring property to the lattice of ideals. In the process, we introduce the concept of an ideal-symmetric ring. We first characterize the class of ideal-symmetric rings and show that this ideal-symmetric property is Morita invariant. We provide a method of constructing an ideal-symmetric ring (but not semiprime) from any given semiprime ring, noting that semiprime rings are ideal-symmetric. We investigate the structure of minimal ideal-symmetric rings completely, finding two kinds of basic forms of finite ideal-symmetric rings. It is also shown that the ideal-symmetric property can go up to right quotient rings in relation with regular elements. The polynomial ring R[x] over an ideal-symmetric ring R need not be ideal-symmetric, but it is shown that the factor ring R[x]/xⁿR[x] is ideal-symmetric over a semiprime ring R.     

Remarks on Centers of Rings

It is proved that for matrices A,B in the n by n upper triangular matrix ring Tn(R) over a domain R,if AB is nonzero and central in Tn(R) then AB =BA.The n by n full matrix rings over right Noetherian domains are also shown to have this property.In this article we treat a ring property that is a generalization of this result,and a ring with such a property is said to be weakly reversible-over-center.The class of weakly reversible-over-center rings contains both full matrix rings over right Noetherian domains and upper triangular matrix rings over domains.The structure of various sorts of weakly reversible-over-center rings is studied in relation to the questions raised in the process naturally.We also consider the connection between the property of being weakly reversible-over-center and the related ring properties.     

A Property Satisfying Reducedness over Centers

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated,especially related to polynomial rings.It is proved that for pseudo-reduced-over-center rings of nonzero characteristic,the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals.For a locally finite ring R,it is proved that if R is pseudo-reduced-over-center,then R is commutative and R/J(R) is a commutative regular ring with J(R) nil,where J(R) is the Jacobson radical of R.     

Bounded Factorization Rings

In this article we present some results about bounded factorization rings (BFRs), i.e., commutative rings with the property that each nonzero nonunit has a bound on the length of its factorizations into nonunits. In their article Factorization in Commutative Rings with Zero Divisors, Anderson and Valdes-Leon conjectured that R[x], the polynomial ring over R, is a bounded factorization ring if and only if R is a BFR and 0 is primary in R. We give some conditions under which the conjecture is true and present a bounded factorization ring with 0 primary where the polynomial ring is not a BFR.     

每一个元素都是左零因子之环

本文引进左(右)零因子环的概念,它们是一类无单位元的环.我们称一个环为左(右)零因子环,如果对于任何 $a \in R$,都有$r_R (a) \neq 0~(l_R(a)\neq 0)$,而称一个环为强左(右)零因子环,如果$r_R(R)\neq 0~(l_R(R)\neq 0)$.Camillo和Nielson称一个环$R$为右有限零化环(简称RFA-环),如果$R$的每一个有限子集都有非零的右零化子.本文给出左零因子环的一些基本例子,探讨强左零因子环和RFA-环的扩张,并给出它们的等价刻画.     

第n根性质与根环

令E为 (MR)的自同态环 ,本文首先证明了MR 有第n根性质的充要条件是EE 有第n根性质 ,由此引进了根环的概念 ,并研究了根环的性质。特别地我们证明了如nA nB且A拟内射 ,则A B .     

The unique rank property among gabriel localizations

It is not uncommon for rings to have Gabriel localizations which do not possess the unique rank (UR) property although the rings themselves do have UR. We show that if F is a Gabriel filter of right ideals on a ring R and R_F is the corresponding Gabriel localization, then free R_F?modules of ranks m and n are isomorphic if and only if some F-dense submodule of (R/T_f(R))^m is isomorphic to some F-dense submodule of (R/T_F(R))ⁿ, where T_F(R) is the F-torsion ideal of R.