Jiaqun Wei

(n, t)-Quasi-Projective and Equivalences

ABSTRACT We introduce the notion of (n, t)-quasi projective and show that they are intimately relative to equivalences of module categories. As applications, we give a classification of * n -modules and generalize Fullers quasi-progenerators naturally.

ω-Gorenstein Modules

We introduce the notion of ω-Gorenstein modules, where ω is a faithfully balanced self-orthogonal module. This gives a common generalization of both Gorenstein projective modules and Gorenstein injective modules. We consider such modules in the tilting theory. Consequently, some results due to Auslander and colleagues and Enochs and colleagues are generalized.

Power-Substitution and Exchange Rings

We proved that an exchange ring R the power-substitution property if and only if any one of the following conditions hold: (1) whenever x ∈ R is regular, there exists some positive integer n such that xI n = xWx for some unit-regular element W ∈ M n (R); (2) whenever x ∈ R is regular, there exist positive integers m, n such that x m I n = x m Wx m for some unit-regular element W ∈ M n (R); (3) whenever x = xyx in R, there exists some positive integer n such that xI n = xyW = Wyx for some unit-regular element W ∈ M n (R); (4) whenever aR + bR = dR in R, there exist some positive integer n and W, Q ∈ M n (R), where W is unit-regular, such that aI n + bQ = dW; (5) whenever a 1 R +···+ a k R = dR in R, where k ≥ 1, there exist some positive integer n and unit-regular elements W 1, …, W k ∈ M n (R) such that a 1 W 1 +···+ a k W k = dI n ; (6) whenever a 1 R +···+ a k R = dR in R, where k ≥ 1, there exist positive integers m, n and unit-regular elements W 1, …, W k ∈ M n (R) such that . These results, by replacing the word “unit” with the word “unit-regular, ” generalize the corresponding results of Canfell, Chen, Wu, etc.

Quasi-cotilting modules and hereditary quasi-cotilting modules

ABSTRACT In this paper, we firstly give some basic properties on quasi-cotilting modules. With the help of these properties, we obtain a Quasi-cotilting Theorem (see Theorem 2.8). In particular, the Cotilting Theorem in [5] is a corollary of our result. Finally, we introduce a new notion, hereditary quasi-cotilting modules, and mainly prove that the class consisting of all Δ-reflexive modules for a hereditary quasi-cotilting module is closed under submodules.

On Balance for Relative Homology

Let R be a ring. It is reasonable to use a strict 𝒲𝒳-resolution of a right R-module M of finite 𝒳-projective dimension, where 𝒳 denotes a subcategory of right R-modules closed under extensions and admits an injective cogenerator 𝒲, to define the relative homology functor . A general balance result is established for such relative homology functor that encompasses a theorem of Emmanouil on Gorenstein projective R-modules and extends a theorem of Holm on Gorenstein flat R-modules. We also consider the above relative homology functor with respect to subcategories arising from an arbitrary but fixed semidualizing R-module. Inspired by the idea of Sather-Wagstaff et al. [25], we obtain the corresponding balance results when R is Cohen–Macaulay with a dualizing R-module as applications of our balance result.

Igusa–Todorov Endomorphism Algebras

Let A be an artin algebra and A V be a 2-subgenerator such that End A V is Ω3-representation-finite. We prove that, for any M ∈ add A V, the endomorphism algebra of M is an Igusa-Todorov algebra and hence its finitistic dimension is finite. The result gives a common generalization of [11, Corollaries 8 and 9], [13, Theorem 1.1] and [16, Theorems 2.3 and 2.5]. We deduce many corollaries and show that Ω2-representation-finiteness is closed under taking endomorphism algebras of projective modules.

n-C-Star Modules and n-C-Tilting Modules

Let C be a faithfully balanced selforthogonal module over an Artin algebra R. We introduce the notion of n-C-star modules, which is a common generalization of n-star modules and n-C-tilting modules. We extend some characterizations of n-star modules to this context and prove that n-C-tilting modules are precisely n-C-star modules n-C-presenting all the injectives.

On a Question of Kilp and Knauer

Abstract Let S be a monoid. We proved the following theorem: all strongly flat left S-acts are regular if and only if all finitely generated strongly flat left S-acts are regular if and only if all cyclic strongly flat left S-acts are regular if and only if S is left semiperfect and a left PP monoid. This gives a complete answer to a question posed by Kilp and Knauer [Kilp, M., Knauer, U. (1987). Characterization of monoids by properties of regular acts. J. Pure Appl. Algebra 46:217–231].

Wakamatsu-silting complexes

ABSTRACT We introduce Wakamatsu-silting complexes (resp., Wakamatsu-tilting complexes) as a common generalization of both silting complexes (resp., tilting complexes) and Wakamatsu-tilting modules. Characterizations of Wakamatsu-silting complexes are given. In particular, we show that a complex T is Wakamatsu-silting if and only if its dual DT is Wakamatsu-silting. It is conjectured that compact Wakamatsu-silting complexes are just silting complexes. We prove that the conjecture lies under the finitistic dimension conjecture.

Unimodular Regularity, Stable Range, and Exchange Rings

Let R be an exchange ring. In this article, we show that the following conditions are equivalent: (1) R has stable range not more than n; (2) whenever x ∈ R n is regular, there exists some unimodular regular w ∈ n R such that x = xwx; (3) whenever x ∈ R n is regular, there exist some idempotent e ∈ R and some unimodular regular w ∈ R n such that x = ew; (4) whenever x ∈ R n is regular, there exist some idempotent e ∈ M n (R) and some unimodular regular w ∈ R n such that x = we; (5) whenever a( n R) + bR = dR with a ∈ R n and b,d ∈ R, there exist some z ∈ R n and some unimodular regular w ∈ R n such that a + bz = dw; (6) whenever x = xyx with x ∈ R n and y ∈ n R, there exist some u ∈ R n and v ∈ n R such that vxyu = yx and uv = 1. These, by replacing unimodularity with unimodular regularity, generalize the corresponding results of Canfell (1995, Theorem 2.9), Chen (Chen 2000, Theorem 4.2 and Proposition 4.6, Chen 2001, Theorem 10), and Wu and Xu (1997, Theorem 9), etc.