Xiaosheng Zhu

Strict Mittag–Leffler Conditions on Gorenstein Modules

In this article, we investigate the relations between Gorenstein projective modules and Gorenstein flat modules in terms of strict Mittag–Leffler condition. We give some conditions under which Gorenstein projectives are Gorenstein flat, and discuss when the direct limits of Gorenstein projective modules are Gorenstein flat. Moreover, we study the dual of Gorenstein injective modules with strict Mittag–Leffler condition.

Cotorsion Theories Involving Auslander and Bass Classes

Let R be a commutative ring, 𝒞 be a semidualizing R-module. We show that the Auslander class 𝒜𝒞(R) with respect to 𝒞 is the first left orthogonal class of some pure injective module M, that is, 𝒜𝒞(R) =⊥1 M, and the Bass class ℬ𝒞(R) is the first right orthogonal class of some G 𝒞-projective module N, that is, ℬ𝒞(R) = N ⊥1 . As applications, we can see that (𝒜𝒞(R), 𝒜𝒞(R)⊥) is a cotorsion theory generated by a set. Especially, we show that (⊥ℬ𝒞(R), ℬ𝒞(R)) is a complete hereditary cotorsion theory cogenerated by a set.

The Vanishing of Connecting Maps in Algebraic K-Theory

In this article, we mainly consider the problem of the vanishing of connecting maps in K-theory. We obtain some conditions under which the connecting map vanishes. Moreover, we give some applications of our results to the field of C*-algebras.

Lifting Central Idempotents Modulo Jacobson Radicals and Ranks of K 0 Groups

Let R be a ring and let J(R) be the Jacobson radical of R. We discuss the problem of determining when the central idempotents in R/J(R) can be lifted to R. If R is a noetherian (artinian) ring, we give some conditions relative to the ranks of K 0 groups under which the central idempotents in R/J(R) can be lifted. In particular, when R is semilocal, these conditions are necessary and sufficient. Moreover, we consider ranks of K 0 groups of pullbacks of rings and obtain the upper and lower bounds on them under some suitable conditions.

Convexity of Torsion Subgroups in K 0 Groups and Dimension Group Properties of K 0 of Pullbacks

Firstly, we characterize the partially ordered K 0 groups of some rings. Secondly, let R be a ring, we discuss the problem when the pre-order on K 0(R) is actually a partial order and when Tor(K 0(R)) is a convex subgroup of K 0(R). Finally, we examine the transfer of some ordering properties (such as partial order, unperforated, interpolation property) on K 0 groups of rings to the K 0 groups of pullbacks. Let R be a pullback of R 1 and R 2 over S, under some suitable conditions, we prove that if each K 0(R i ) (i = 1, 2) is a dimension group, then so is K 0(R).

Ranks of Low K-Groups of Fibre Products

In this article, we discuss the ranks of low K-groups in pullback diagrams and give the upper and lower bounds on ranks of K-groups of a fibre product (or pullback) under some suitable conditions.