Related papers: On a recent generalization of semiperfect rings
Given a commutative ring $R$ and finitely generated ideal $I$, one can consider the classes of $I$-adically complete, $L_0^I$-complete and derived $I$-complete complexes. Under a mild assumption on the ideal $I$ called weak pro-regularity,…
A new class of rings, the class of left localizable rings, is introduced. A ring $R$ is left localizable if each nonzero element of $R$ is invertible in some left localization $S^{-1}R$ of the ring $R$. Explicit criteria are given for a…
We prove that every isometry of between (not-necessarily orthogonal) summands of a unimodular quadratic space over a semiperfect ring can be extended an isometry of the whole quadratic space. The same result was proved by Reiter for the…
We first introduce and study the notion of semi-regular flat modules, and then show that a ring $R$ is a strong \Prufer\ ring if and only if every submodule of a semi-regular flat $R$-module is semi-regular flat, if and only if every ideal…
Flat modules play an important role in the study of the category of modules over rings and in the characterization of some classes of rings. We study the e-flatness for semimodules introduced by the first author using his new notion of…
Let $R$ be a commutative ring. An $R$-module $M$ is called a semi-regular $w$-flat module if $\Tor_1^R(R/I,M)$ is $\GV$-torsion for any finitely generated semi-regular ideal $I$. In this article, we show that the class of semi-regular…
We define generalized Koszul modules and rings and develop a generalized Koszul theory for $\mathbb{N}$-graded rings with the degree zero part noetherian semiperfect. This theory specializes to the classical Koszul theory for graded rings…
Tilting modules over commutative rings were recently classified in [12]: they correspond bijectively to faithful Gabriel topologies of finite type. In this note we extend this classification by dropping faithfulness. The counterpart of an…
We introduce a complete radical formula for modules over non-commutative rings which is the equivalence of a radical formula in the setting of modules defined over commutative rings. This gives a general frame work through which known…
For a fixed ring, different classes of ring epimorphisms and localisation maps are compared. In fact, we provide sufficient conditions for a ring epimorphism to be a universal localisation. Furthermore, we consider recollements induced by…
We introduce and study a nontrivial generalization of uniserial modules and rings. A module is called weakly uniserial if its submodules are comparable regarding embedding. Also, a right (resp., left) weakly uniserial ring is a ring which…
Based upon properties of ordinal length, we introduce a new class of modules, the binary modules, and study their endomorphism ring. The nilpotent endomorphisms form a two-sided ideal, and after factoring this out, we get a commutative…
In this paper, we construct indecomposable integrally closed modules of arbitrary rank over a two-dimensional regular local ring. The modules are quite explicitly constructed from a given complete monomial ideal. We also give structural and…
We introduce a notion of generalized homogeneous derivations on graded rings as a natural extension of the homogeneous derivations defined by Kanunnikov. We then define gr-generalized derivations, which preserve the degrees of homogeneous…
Given a ring $R$, we study the bimodules $M$ for which the trivial extension $R\propto M$ is morphic. We obtain a complete characterization in the case where $R$ is left perfect, and we prove that $R\propto Q/R$ is morphic when $R$ is a…
Inspired by Jorgensen et. al., it is proved that if a Cohen--Macaulay local ring $R$ with dualizing module admits a suitable chain of semidualizing $R$--modules of length $n$, then $R\cong Q/(I_1+\cdots+I_n)$ for some Gorenstein ring $Q$…
In this paper we consider the question of when the associated graded ring along a valuation, ${\rm gr}_{\nu^*}(S)$, is a finite ${\rm gr}_{\nu^*}(R)$-module, where $S$ is a normal local ring which lies over a normal local ring $R$ and…
We develop the theory of central ideals on commutative rings. We introduce and study the central seminormalization of a ring in another one. This seminormalization is related to the theory of regulous functions on real algebraic varieties.…
A submodule $W$ of $V$ is summand absorbing, if $x + y \in W$ implies $x \in W, \; y \in W $ for any $x, y \in V$. Such submodules often appear in modules over (additively) idempotent semirings, particularly in tropical algebra. This paper…
For a commutative ring $A$, we have the category of (bounded-below) chain complexes of $A$-modules $Ch_{+}(A\mymod)$, a closed symmetric monoidal category with a compatible stable Quillen model structure. The associated homotopy category is…