$PS$-Injective Modules, $PS$-Flat Modules and $PS$-Coherent Rings

A left ideal $I$ of a ring $R$ is small in case for every proper left ideal $K$ of $R, K +I≠R$. A ring $R$ is called left $PS$-coherent if every principally small left ideal $Ra$ is finitely presented. We develop, in this paper, $PS$-coherent rings as a generalization of $P$-coherent rings and $J$-coherent rings. To characterize $PS$-coherent rings, we first introduce $PS$-injective and $PS$-flat modules, and discuss the relation between them over some spacial rings. Some properties of left $PS$-coherent rings are also studied.     

幺半群上的斜强可逆环

For a monoid M, we introduce the concept of skew strongly M-reversible rings which is a generalization of strongly M-reversible rings, and investigate their properties. It is shown that if G is a finitely generated Abelian group, then G is torsion-free if and only if there exists a ring R with |R| ≥ 2 such that R is skew strongly G-reversible. Moreover, we prove that if R is a right Ore ring with classical right quotient ring Q, then R is skew strongly M-reversible if and only if Q is skew strongly M-reversible.     

关于$\pi$-半交换环

A ring R is said to be π-semicommutative if a, b ∈ R satisfy ab = 0 then there exists a positive integer n such that anRbn= 0. We study the properties of π-semicommutative rings and the relationship between such rings and other related rings. In particular, we answer a question on left GWZI rings negatively.     

弱$r$-Clean环

As generalization of r-clean rings and weakly clean rings, we define a ring R is weakly r-clean if for any a∈R there exist an idempotent e and a regular element r such that a = r + e or a = r-e. Some properties and examples of weakly r-clean rings are given. Furthermore, we prove the weakly clean rings and weakly r-clean rings are equivalent for abelian rings.     

半交换环的一个推广研究

In this paper, a generalization of the class of semicommutative rings is investigated.A ring R is called left GWZI if for any a ∈ R, l(a) is a GW-ideal of R. We prove that a ring R is left GWZI if and only if S3(R) is left GWZI if and only if Vn(R) is left GWZI for any n ≥ 2.     

$\mathcal{F}$-Perfect Rings and Modules

Let $R$ be a ring, and let $(\mathcal{F}, C)$ be a cotorsion theory. In this article, the notion of $\mathcal{F}$-perfect rings is introduced as a nontrial generalization of perfect rings and A-perfect rings. A ring $R$ is said to be right $\mathcal{F}$-perfect if $F$ is projective relative to $R$ for any $F ∈ \mathcal{F}$. We give some characterizations of $\mathcal{F}$-perfect rings. For example, we show that a ring $R$ is right $\mathcal{F}$-perfect if and only if $\mathcal{F}$-covers of finitely generated modules are projective. Moreover, we define $\mathcal{F}$-perfect modules and investigate some properties of them.     

拟-中心半交换环

环$R$称为拟-中心半交换的(简称QCS环)如果对$a,b\in R$, $ab=0$蕴含$aRb\subseteq Q(R)$, 其中$Q(R)$为$R$的拟中心.证明了如果$R$ 为QCS环, 那么$R$的幂零元集恰好是它的Wedderburn根, 且对$n\geq 2$, 上三角矩阵环$R=T_n(S)$ 是QCS 环当且仅当$n=2$ 且$S$ 是duo 环, 而$T_{2k+2}^k$是QCS环如果$R$是约化的duo环.     

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.     

关于斜McCoy环

For a ring endomorphism α,we introduce α-skew McCoy rings which are generalizations of α-rigid rings and McCoy rings,and investigate their properties.We show that if α t = I R for some positive integer t and R is an α-skew McCoy ring,then the skew polynomial ring R[x;α] is α-skew McCoy.We also prove that if α（1） = 1 and R is α-rigid,then R[x;α]/ x 2 is αˉ-skew McCoy.     

广义IP-内射环

对环R,令ip(R_R)={a∈R：任意一个从R的右理想到R且象为aR的模同态能开拓到R}。众所周知,R为右IP-内射环当且仅当R=ip(R_R),R为右单-内射环当且仅当{a∈R：aR is simple)(?)ip(R_R)。对环R的一个子集S,我们引进了S-IP-内射环的概念,即满足S(?)ip(R_R)的环。并得到了这种环的一些性质。     

相对于幺半群的McCoy环的扩张

对于幺半群~$M$, 本文引入了~$M$-McCoy~环.~证明了~$R$~是~$M$-McCoy~环当且仅当~$R$~上的~$n$~阶上三角矩阵环~$aUT_n(R)$~是~$M$-McCoy~环;得到了若~$R$~是~McCoy~环,~$R[x]$~是~$M$-McCoy~环,则~$R[M]$~是~McCoy~环;对于包含无限循环子半群的交换可消幺半群~$M$,证明了若~$R$~是~$M$-McCoy~环,则半群环~$R[M]$~是~McCoy~环及~$R$~上的多项式环~$R[x]$~是~$M$-McCoy~环.     

Some Rings Characterized by Modules

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具有对称自同态的环及其扩张

Let R be a ring with an endomorphism α and an α-derivation δ. We introduce the notions of symmetric α-rings and weak symmetric α-rings which are generalizations of symmetric rings and weak symmetric rings, respectively, discuss the relations between symmetricα-rings and related rings and investigate their extensions. We prove that if R is a reduced ring and α(1) = 1, then R is a symmetric α-ring if and only if R[x]/(x n) is a symmetric ˉα-ring for any positive integer n. Moreover, it is proven that if R is a right Ore ring, α an automorphism of R and Q(R) the classical right quotient ring of R, then R is a symmetric α-ring if and only if Q(R) is a symmetric ˉα-ring. Among others we also show that if a ring R is weakly 2-primal and(α, δ)-compatible, then R is a weak symmetric α-ring if and only if the Ore extension R[x; α, δ] of R is a weak symmetric ˉα-ring.     

强$J$-Semiclean环

We in this note introduce a new concept, so called strongly J-semiclean ring, that is a generalization of strongly J-clean rings. We first observe the basic properties of strongly J-semiclean rings, constructing typical examples. We next investigate conditions on a local ring R that imply that the upper triangular matrix ring T_n(R) is a strongly J-semiclean ring. Also,the criteria on strong J-semicleanness of 2 × 2 matrices in terms of a quadratic equation are given. As a consequence, several known results relating to strongly J-clean rings are extended to a more general setting.     

右IF－环及凝聚环的挠理论

本文研究了右ＩＦ－环的性质，证明出环Ｒ是右ＩＦ－环当且仅当Ｒ是左凝聚环，并且是平坦模；由此证明出右ＩＦ－环与左ＧＱＦ－环是等价的，其次应用右ＩＦ－环研究了凝聚环的挠理论性质，证明出凝聚环与Ｔ－凝聚环的关系。     

正则环的投射根

研究了正则环上投射根的性质.证明了正则环的投射根左右对称,且模去投射根的正则环只有零投射根.给出了矩阵环及角落环投射根的计算式,并得到了投射根为零的正则环的一些刻画。最后讨论了投射根为零的正则环在各种环运算下的封闭性和正则环的MP-维数.     

PVMR的一些刻画

In this paper, we study Prufer wmultiplication rings （PVMRs） and give some new characterizations of PVMRs. Moreover, we show that a Marot ring R is a PVMR if and only if every w-ideal of R is complete.     

斜Armendariz矩阵环

Let R be a ring.We show in the paper that the subring Un(R) of the upper triangular matrix ring Tn(R) is α-skew Armendariz if and only if R is α-rigid,also it is maximal in some non α-skew Armendariz rings,where α is a ring endomorphism of R with α(1) = 1.     

Examples of Armendariz Rings

We construct various examples of Armendariz and related rings by reviewing and extending some results concerning the structure of nil(R). In particular, we give some examples of Armendariz rings related to power series rings and an example of an n-Armendariz ring, for all n ≥ 1, which is not Armendariz.