Zhongkui Liu

On Weak Armendariz Rings

We introduce weak Armendariz rings which are a generalization of semicommutative rings and Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak Armendariz if and only if for any n, the n-by-n upper triangular matrix ring T n (R) is weak Armendariz. If R is semicommutative, then it is proven that the polynomial ring R[x] over R and the ring R[x]/(x n ), where (x n ) is the ideal generated by x n and n is a positive integer, are weak Armendariz.

INDECOMPOSABLE, PROJECTIVE, AND FLATS-POSETS

Abstract For a monoid S , a (left) S -act is a nonempty set B together with a mapping S ×B→B sending (s, b) to sb such that S (tb) = lpar;st)b and 1b  = b for all S , t ∈ S and B  ∈ B. Right S -acts A can also be defined, and a tensor product A  ⊗  s B (a set)can be defined that has the customary universal property with respect to balanced maps from A × B into arbitrary sets. Over the past three decades, an extensive theory of flatness properties has been developed (involving free and projective acts, and flat acts of various sorts, defined in terms of when the tensor product functor has certain preservation properties). A recent and complete discussion of this area is contained in the monograph Monoids, Acts and Categories by M. Kilp et al. (New York: Walter de Gruyter, 2000). To date, there have been only a few attempts to generalize this material to ordered monoids acting on partially ordered sets ( S -posets). The present paper is devoted to such a generalization. A unique decomposition theorem for S -posets is given, based on strongly convex, indecomposable S -subposets, and a structure theorem for projective S -posets is given. A criterion for when two elements of the tensor product of S -posets given, which is then applied to investigate several flatness properties.

A NOTE ON PRINCIPALLY QUASI-BAER RINGS

ABSTRACT Let be a ring such that all left semicentral idempotents are central. It is shown that is right p.q.Baer if and only if is right p.q.Baer and any countable family of idempotents in has a generalized join in .

Gorenstein Projective, Injective, and Flat Complexes

Enochs and Jenda [4] gave some characterizations of Gorenstein injective and projective complexes over n-Gorenstein rings. The aim of this article is to generalize these results and to give homological descriptions of the Gorenstein dimensions over arbitrary associative rings.

ON A GENERALIZATION OF SEMICOMMUTATIVE RINGS

We introduce weakly semicommutative rings which are a generalization of semicommutative rings, and give some examples which show that weakly semicommutative rings need not be semicommutative. Also we give some relations between semicommutative rings and weakly semicommutative rings.

Special Properties of Rings of Generalized Power Series

Abstract Let R be a ring and (S, ≤) a strictly ordered monoid. Properties of the ring [[R S,≤]] of generalized power series with coefficients in R and exponents in S are considered in this paper. It is shown that [[R S,≤]] is reduced (2-primal, Dedekind finite, clean, uniquely clean) if and only if R is reduced (2-primal, Dedekind finite, clean, uniquely clean, respectively) under some additional conditions. Also a necessary and sufficient condition is given for rings under which the ring [[R S,≤]] is a reduced left PP-ring.

Ding Projective and Ding Injective Modules

An R-module M is called Ding projective if there exists an exact sequence ⋯ → P1→ P0→ P0→ P1→ ⋯ of projective R-modules with M=Ker(P0→ P1) such that Hom(-,F) leaves the sequence exact whenever F is a flat R-module. In this paper, we develop some basic properties of such modules. Also, properties of Ding injective modules are discussed.

INJECTIVITY OF MODULES OF GENERALIZED INVERSE POLYNOMIALS

Let R be an associative ring with an identity and (S, ≤) a strictly totally ordered monoid, which is also artinian and finitely generated. If R is left noetherian and M is a left R-module, then we show that i.dim[[R S, ≤]]([M S, ≤]) ≤ i.dim R M. In particular, R M is injective if and only if [[R S, ≤]][M S, ≤] is injective.

Model Structures on Categories of Complexes Over Ding-Chen Rings

The so-called Ding–Chen ring is an n-FC ring which is both left and right coherent, and has both left and right self FP-injecitve dimensions at most n for some non-negative integer n. In this article, we introduce the so-called Ding projective, Ding injective complexes, and show that over Ding–Chen rings the homotopy theory on the category of modules can be extended to a homotopy theory on the category of complexes.

Strongly Gorenstein Flat Dimensions of Complexes

In this article, we define and study a notion of strongly Gorenstein flat dimensions for complexes of left modules over associative rings. In particular, we consider the class of homologically bounded below complexes of left R-modules, and show that strongly Gorenstein flat dimension has a nice functorial description. In addition, we will investigate the strongly Gorenstein flat properties of complexes under change of rings. As an application, we study the Tate cohomology theories with respect to ℱ-complete resolutions, where ℱ is the class of all flat modules.