On SAP-rings

平坦半模与冯·诺依曼正则半环

Flat Semimodules & von Neumann Regular Semirings

本文研究半模的e-平坦性及其与其他平坦性概念的关系，并证明每个右（左）半模均e-平坦的减法半环必为冯·诺依曼正则半环。

von Neumann regular Hyperrings and applications to Real Reduced Multirings

A multiring ([Mar3]) is a kind of ring where is allowed the sum of two elements to be anon-empty subset of the structure instead of just one element -and an hyperring is a multiring with a strong distributive property. Thus a reduced hyperring where the prime spec is a Boolean topological space is called von Neumann regular hyperring (vNH). It is possible to associate to every such object a structural presheaf in the same way it is made with rings but there are some vNH such that this presheaf is not a sheaf. In this sense, we give a first-order characterization of vNH with a structural sheaf …

Feedback classification of linear systems over von Neumann regular rings

It is proved that feedback classification of a linear system over a commutative von Neumann regular ring R can be reduced to the classification of a finite family of systems, each of which is properly split into a reachable and a non-reachable part, where the reachable part is in a Brunovski-type canonical form, while the non-reachable part can only be altered by similarity. If a canonical form is known for similarity of matrices over R, then it can be used to construct a canonical form for feedback equivalence. An explicit algorithm is given to obtain the canonical form in a computable contex…

Konstruktion von Invariantenringen ohne die Cohen-Macaulay Eigenschaft

We give examples of Non-Cohen-Macaulay invariant rings.

Über die von einem Ideal $I \subset R$ erzeugten $R$-Moduln III

Let $(R, \mathfrak m)$ be a commutative noetherian local ring and $I$ an ideal of $R$. For every $R$-module $M$, $γ_I(M) = \sum\{ \operatorname{Bi} f \,|\, f \in \operatorname{Hom}_R(I,M)\}$ is called the trace of $I$ in $M$. It is easy to see that $\operatorname{Ext}_R^1(R/I,M) = 0$ always implies $IM = γ_I(M)$. If the second condition holds for all ideals $I$ of $R$, we say that $M$ is excellent. In part 1, we show a number of conditions for these modules, which are well-known for injective modules. In the second part, we examine the special case $M = R$. In particular, we show that for ever…

关于交换环中某些特殊元素的S版本

On the $S$-version of some special elements in commutative rings

本文研究了交换环中若干基本元素的S版本，定义并探讨了S可逆、S幂等、S von Neumann 正则及S-π正则元素的性质，用于刻画环类，并关注布尔环和π-正则环的统一S版本，提供区分其经典类比的例子。

On rings whose finitely generated left ideals are left annihilators of an element

An associative ring $R$ with identity is left pseudo-morphic if for every $a$$\in$$R$, there exists $b$$\in$$R$ such that $Ra=l_R(b)$. If, in addition, $l_R(a)=Rb$, then $R$ is called left morphic. $R$ is morphic if it is both left and right morphic. We characterize left pseudo-morphic rings; identify the cases a (left) pseudo morphic ring is (left) quasi-morphic, morphic, Quasi-Frobenius, von Neumann regular, etc.; correct two results in a book and a paper; and completely determine when the trivial extension of a commutative domain is morphic which positively answered a question in a paper.

Characterizations of regular modules

Different and distinct notions of regularity for modules exist in the literature. When these notions are restricted to commutative rings, they all coincide with the well-known von-Neumann regularity for rings. We give new characterizations of these distinct notions for modules in terms of both (weakly-)morphic modules and reduced modules. Furthermore, module theoretic settings are established where these in general distinct notions turn out to be indistinguishable.

Über die von einem Ideal $I \subset R$ erzeugten $R$-Moduln

Let $(R, \mathfrak m)$ be a commutative noetherian local ring. We investigate under which conditions an $R$-module $M$ is generated by an ideal $I$, i.e. there exists an epimorphism $I^{(Λ)} \twoheadrightarrow M$. If $M$ is uniserial, i.e. $\mathcal{L}(M)$ is totally ordered and finite, this is equivalent to $\mathfrak{m}^{n-1} \cdot I \not\subset \operatorname{Ann}_R(M) \cdot I$ ($\operatorname{length}(M) = n \geq 1$). If $M$ is cyclic and $I = \mathfrak{m}$, this is equivalent to: Either it is $M \cong R/\mathfrak{p}$ ($R/\mathfrak{p}$ a discrete valuation ring) or $M \cong C/\operatorname{S…

交换环的$q$-Krull维数

$q$-Krull dimension of commutative rings

本文引入并研究交换环的$q$-Krull维数，得到$τ_q$-von Neumann正则环新刻画，探究维数为0的环的性质，刻画$τ_q$-Noetherian环的该维数，还正面回答了H. Kim提出的相关问题。

关于 annihilator 乘法模

On Annihilator Multiplication Modules

本文研究 annihilator 乘法模的代数性质，提出新特征化方法，并通过平凡扩张构造非乘法环的例子，证明 Ass_A(E)=Ass(A) 的等价性。

On $τ_q$-flatness and $τ_q$-coherence

In this paper, we introduce and study the notions of $τ_q$-flat modules and $τ_q$-coheret rings. First, by investigating the Nagata rings of $τ_q$-torsion theory, we show that the small finitistic dimensions of T$(R[x])$ are all equal to $0$ for any ring $R$. Then, we introduce the notion of $τ_q$-VN regular rings (i.e. over which all modules are $τ_q$-flat), and show that a ring $R$ is a $τ_q$-VN regular ring if and only if T$(R[x])$ is a von Neumann regular ring. Finally, we obtain the Chase theorem for $τ_q$-coheret rings: a ring $R$ is $τ_q$-coherent if and only if any direct product of $R…

关于均匀 $S$-版本环、模及其同调理论的综述

A survey on the uniform $S$-version of rings, modules and their homological theories

本文综述了均匀 $S$-版本环、模及其同调理论的最新进展，介绍了 $u$-$S$ 模类及其相关同调维数。