Gaohua Tang

Self-orthogonal modules over coherent rings

Abstract Let R be a left coherent ring, S any ring and R ω S an ( R , S )-bimodule. Suppose ω S has an ultimately closed FP-injective resolution and R ω S satisfies the conditions: (1) ω S is finitely presented; (2) The natural map R → End( ω S ) is an isomorphism; (3) Ext S i ( ω , ω )=0 for any i ≥1. Then a finitely presented left R -module A satisfying Ext R i ( A , ω )=0 for any i ≥1 implies that A is ω -reflexive. Let R be a left coherent ring, S a right coherent ring and R ω S a faithfully balanced self-orthogonal bimodule and n ≥0. Then the FP-injective dimension of R ω S is equal to or less than n as both left R -module and right S -module if and only if every finitely presented left R -module and every finitely presented right S -module have finite generalized Gorenstein dimension at most n .

Study of Morita contexts

This article concerns mainly on various ring properties of Morita contexts. Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a generalized matrix ring over a ring to satisfy a certain ring property which is among being semilocal, semiperfect, left perfect, semiprimary, semipotent, potent, clean, strongly π-regular, semiregular, etc. Many known results on a formal triangular matrix ring are extended to a Morita context or a trivial Morita context. Some questions on this subject raised by Varadarajan in [22] are answered.

Commutative Zero-divisor Semigroups of Graphs with at Most Four Vertices

The zero-divisor graph of a commutative semigroup with zero is a graph whose vertices are the nonzero zero-divisors of the semigroup, with two distinct vertices joined by an edge in case their product in the semigroup is zero. In this paper, we study commutative zero-divisor semigroups determined by graphs. We determine all corresponding zero-divisor semigroups of all simple graphs with at most four vertices.

The Square Mapping Graphs of Finite Commutative Rings

For a finite commutative ring R, the square mapping graph of R is a directed graph Γ(R) whose set of vertices is all the elements of R and for which there is a directed edge from a to b if and only if a2=b. We establish necessary and sufficient conditions for the existence of isolated fixed points, and the cycles with length greater than 1 in Γ(R). We also examine when the induced subgraph on the set of zero-divisors of a local ring with odd characteristic is semiregular. Moreover, we completely determine the finite commutative rings whose square mapping graphs have exactly two, three or four components.

Quasipolar Property of Generalized Matrix Rings

The article concerns the question of when a generalized matrix ring K s (R) over a local ring R is quasipolar. For a commutative local ring R, it is proved that K s (R) is quasipolar if and only if it is strongly clean. For a general local ring R, some partial answers to the question are obtained. There exist noncommutative local rings R such that K s (R) is strongly clean, but not quasipolar. Necessary and sufficient conditions for a single matrix of K s (R) (where R is a commutative local ring) to be quasipolar is obtained. The known results on this subject in [5] are improved or extended.

CODIMENSION AND REGULARITY OVER COHERENT SEMILOCAL RINGS

In this paper, the results on codimension and regularity over noetherian local rings and coherent local rings are extended to coherent semilocal rings and some useful examples of coherent semilocal rings are constructured.

Matrices over a commutative ring as sums of three idempotents or three involutions

Abstract Motivated by Hirano-Tominaga’s work on rings for which every element is a sum of two idempotents and by de Seguins Pazzis’s results on decomposing every matrix over a field of positive characteristic as a sum of idempotent matrices, we address decomposing every matrix over a commutative ring as a sum of three idempotent matrices and, respectively, as a sum of three involutive matrices.

Nil-clean group rings

An element a of a ring R is nil-clean, if a = e + b, where e2 = e ∈ R and b is a nilpotent element, and the ring R is called nil-clean if each of its elements is nil-clean. In [W. Wm. McGovern, S. Raja and A. Sharp, Commutative nil clean group rings, J. Algebra Appl. 14(6) (2015) 5; Article ID: 1550094], it was proved that, for a commutative ring R and an abelian group G, the group ring RG is nil-clean, iff R is nil-clean and G is a 2-group. Here, we discuss the nil-cleanness of group rings in general situation. We prove that the group ring of a locally finite 2-group over a nil-clean ring is nil-clean, and that the hypercenter of the group G must be a 2-group if a group ring of G is nil-clean. Consequently, the group ring of a nilpotent group over an arbitrary ring is nil-clean, iff the ring is a nil-clean ring and the group is a 2-group.

On *-clean non-commutative group rings

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In this paper, we consider the group algebras of the dihedral groups D2n, and the generalized quaternion groups Q2n with standard involution ∗. For the non-semisimple group algebra case, we characterize the ∗-cleanness of RD2pk with a prime p ∈ J(R), and RD2n with 2 ∈ J(R), where R is a commutative local ring. For the semisimple group algebra case, we investigate when KG is ∗-clean, where K is the field of rational numbers ℚ or a finite field Fq and G = D2n or G = Q2n.

ON φ-VON NEUMANN REGULAR RINGS

Let R be a commutative ring with and let = {R|R is a commutative ring and Nil(R) is a divided prime ideal}. If , then R is called a -ring. In this paper, we introduce the concepts of -torsion modules, -flat modules, and -von Neumann regular rings.