拓扑半完全拓扑环

Semi-Invariant Subrings

We say that a subring $R_0$ of a ring $R$ is semi-invariant if $R_0$ is the ring of invariants in $R$ under some set of ring endomorphisms of some ring containing $R$. We show that $R_0$ is semi-invariant if and only if there is a ring $S\supseteq R$ and a set $X\subseteq S$ such that $R_0=\Cent_R(X):={r\in R \suchthat xr=rx \forall x\in X}$; in particular, centralizers of subsets of $R$ are semi-invariant subrings. We prove various properties of semi-invariant subrings and show how they can be used for various applications including: (1) The center of a semiprimary (resp. right perfect) ring …

单模的自同态环与块分解

Endomorphism rings of simple modules and block decomposition

本文利用左右诺特半完全环不可分解性的特征刻画，通过Ext群研究单模，证明不可分解半完全环的单模基数及自同态环特征的一致性，并在拟Frobenius环情形下强化了结果。

Homological dimensions of the Jacobson radical

This work presents results on the finiteness, and on the symmetry properties, of various homological dimensions associated to the Jacobson radical and its higher syzygies, of a semiperfect ring.

Topologically semisimple and topologically perfect topological rings

Extending the Wedderburn-Artin theory of (classically) semisimple associative rings to the realm of topological rings with right linear topology, we show that the abelian category of left contramodules over such a ring is split (equivalently, semisimple) if and only if the abelian category of discrete right modules over the same ring is split (equivalently, semisimple). Our results in this direction complement those of Iovanov-Mesyan-Reyes. An extension of the Bass theory of left perfect rings to the topological realm is formulated as a list of conjecturally equivalent conditions, many equival…

Rings over which every module has a flat $δ$-cover

Let $M$ be a module. A {\em $δ$-cover} of $M$ is an epimorphism from a module $F$ onto $M$ with a $δ$-small kernel. A $δ$-cover is said to be a {\em flat $δ$-cover} in case $F$ is a flat module. In the present paper, we investigate some properties of (flat) $δ$-covers and flat modules having a projective $δ$-cover. Moreover, we study rings over which every module has a flat $δ$-cover and call them {\em right generalized $δ$-perfect} rings. We also give some characterizations of $δ$-semiperfect and $δ$-perfect rings in terms of locally (finitely, quasi-, direct-) projective $δ$-covers and flat …

On Semilocal Modules and Rings

It is well-known that a ring R is semiperfect if and only if R as a left (or as a right) R-module is a supplemented module. Considering weak supplements instead of supplements we show that weakly supplemented modules M are semilocal (i.e., M/Rad(M) is semisimple) and that R is a semilocal ring if and only if R as a left (or as a right) R-module is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie dimension) of modules is of interest and yields a natural interpretation of the Camps-Dicks characterization of semilocal rings. Finitely generated modu…

A generalization of supplemented modules

Let $M$ be a left module over a ring $R$ and $I$ an ideal of $R$. $M$ is called an $I$-supplemented module (finitely $I$-supplemented module) if for every submodule (finitely generated submodule) $X$ of $M$, there is a submodule $Y$ of $M$ such that $X+Y=M$, $X\cap Y\subseteq IY$ and $X\cap Y$ is PSD in $Y$. This definition generalizes supplemented modules and $δ$-supplemented modules. We characterize $I$-semiregular, $I$-semiperfect and $I$-perfect rings which are defined by Yousif and Zhou [15] using $I$-supplemented modules. Some well known results are obtained as corollaries.

ADS modules

We study the class of ADS rings and modules introduced by Fuchs. We give some connections between this notion and classical notions such as injectivity and quasi-continuity. A simple ring R such that R is ADS as a right R-module must be either right self-injective or indecomposable as a right R-module. Under certain conditions we can construct a unique ADS hull up to isomorphism. We introduce the concept of completely ADS modules and characterize completely ADS semiperfect right modules as direct sum of semisimple and local modules.

由silting模生成的TTF类

TTF classes generated by silting modules

研究环上模范畴中TTF类成为silting类的条件，通过幂等理想与TTF类的对应关系，得出R/I为silting模的充要条件，证明I是投射模迹时R/I为silting模且逆命题对半完全环等成立。

局部环上的弦代数：容许性与双列性

String algebras over local rings: admissibility and biseriality

本文定义局部环上的弦代数，给出容许理想的充分条件，证明其双列性并描述根中单列项的合冲，同时提供离散赋值环上的 Bäckström 序例子。

相对大投射模及其在直和分解中的应用

Relatively big projective modules and their applications to direct sum decompositions

本文扩展了可数生成投射模相对于迹理想总是相对大的环的例子列表，证明其包括满足多项式恒等式的右noetherian环及$h$-局部整环上的局部半完备无挠代数，并以此完整刻画了Krull维数为1的$h$-局部整环上有限生成无挠模直和项的分解性质。

Contramodules over pro-perfect topological rings

For four wide classes of topological rings $\mathfrak R$, we show that all flat left $\mathfrak R$-contramodules have projective covers if and only if all flat left $\mathfrak R$-contramodules are projective if and only if all left $\mathfrak R$-contramodules have projective covers if and only if all descending chains of cyclic discrete right $\mathfrak R$-modules terminate if and only if all the discrete quotient rings of $\mathfrak R$ are left perfect. Three classes of topological rings for which this holds are the complete, separated topological associative rings with a base of neighborhood…

Diagonal-preserving ring *-isomorphisms of Leavitt path algebras

The graph groupoids of directed graphs are topologically isomorphic if and only if there is a diagonal-preserving ring *-isomorphism between the Leavitt path algebras.

On lambda-rings and topological realization

It is shown that most possibly truncated power series rings admit uncountably many filtered lambda-ring structures. The question of how many of these filtered lambda-ring structures are topologically realizable by the K-theory of torsionfree spaces is also considered for truncated polynomial rings.