Yiqiang Zhou

RINGS IN WHICH ELEMENTS ARE UNIQUELY THE SUM OF AN IDEMPOTENT AND A UNIT

An associative ring with unity is called clean if every element is the sum of an idempotent and a unit; if this representation is unique for every element, we call the ring uniquely clean. These rings represent a natural generalization of the boolean rings in that a ring is uniquely clean if and only if it is boolean modulo the Jacobson radical and idempotents lift uniquely modulo the radical. We also show that every image of a uniquely clean ring is uniquely clean, and construct several noncommutative examples.

Armendariz and Reduced Rings

Abstract A ring R is called Armendariz if, whenever in R[x], a i b j = 0 for all i and j. In this paper, some “relatively maximal” Armendariz subrings of matrix rings are identified, and a necessary and sufficient condition for a trivial extension to be Armendariz is obtained. Consequently, new families of Armendariz rings are presented.

On Strongly Clean Matrix and Triangular Matrix Rings

A ring R with identity is called “clean” if every element of R is the sum of an idempotent and a unit, and R is called “strongly clean” if every element of R is the sum of an idempotent and a unit that commute. Strongly clean rings are “additive analogs” of strongly regular rings, where a ring R is strongly regular if every element of R is the product of an idempotent and a unit that commute. Strongly clean rings were introduced in Nicholson (1999) where their connection with strongly π-regular rings and hence to Fittings Lemma were discussed. Local rings and strongly π-regular rings are all strongly clean. In this article, we identify new families of strongly clean rings through matrix rings and triangular matrix rings. For instance, it is proven that the 2 × 2 matrix ring over the ring of p-adic integers and the triangular matrix ring over a commutative semiperfect ring are all strongly clean.

MODULES WHICH ARE INVARIANT UNDER AUTOMORPHISMS OF THEIR INJECTIVE HULLS

A module is defined to be an automorphism-invariant module if it is invariant under automorphisms of its injective hull. Quasi-injective modules and, more generally, pseudo-injective modules are all automorphism-invariant. Here we develop basic properties of these modules, and discuss when an automorphism-invariant module is quasi-injective or injective. Some known results on quasi-injective and pseudo-injective modules are extended.

Classes of Modules

PRELIMINARY BACKGROUND Notation and Terminology Lattices IMPORTANT MODULE CLASSES AND CONSTRUCTIONS Torsion Theory The Module Class s[M] Natural Classes M-Natural Classes Pre-Natural Classes FINITENESS CONDITIONS Ascending Chain Conditions Descending Chain Conditions Covers and Ascending Chain Conditions TYPE THEORY OF MODULES: DIMENSION Type Submodules and Type Dimensions Several Type Dimension Formulas Some Non-Classical Finiteness Conditions TYPE THEORY OF MODULES: DECOMPOSITIONS Type Direct Sum Decompositions Decomposability of Modules Unique Type Closure Modules TS-Modules LATTICES OF MODULE CLASSES The Lattice of Pre-Natural Classes More Sublattice Structures Lattice Properties of Npr (R) More Lattice Properties of Npr (R) The Lattice Nr(R) and Its Applications The Boolean Ideal Lattice REFERENCES INDEX

GP-Injective Rings Need Not be P-Injective

Abstract A ring R is called left P-injective if for every a ∈ R, aR = r(l(a)) where l( ⋅ ) and r( ⋅ ) denote left and right annihilators respectively. The ring R is called left GP-injective if for any 0 ≠ a ∈ R, there exists n > 0 such that a n ≠ 0 and a n R = r(l(a n )). As a response to an open question on GP -injective rings, an example of a left GP-injective ring which is not left P-injective is given. It is also proved here that a ring R is left FP -injective if and only if every matrix ring 𝕄 n (R) is left GP-injective.

STRONGLY CLEAN POWER SERIES RINGS

An element a in a ring R with identity is called strongly clean if it is the sum of an idempotent and a unit that commute. And a ∈ R is called strongly π-regular if both chains aR ⊇ a2R ⊇ · · · and Ra ⊇ Ra2 ⊇ · · · terminate. A ring R is called strongly clean (respectively, strongly π-regular) if every element of R is strongly clean (respectively, strongly π-regular). Strongly π-regular elements of a ring are all strongly clean. Let σ be an endomorphism of R. It is proved that for Σrix ∈ R[x, σ] , if r0 or 1−r0 is strongly π-regular in R, then Σrix is strongly clean in R[x, σ] . In particular, if R is strongly π-regular, then R[x, σ] is strongly clean. It is also proved that if R is a strongly π-regular ring, then R[x, σ]/(xn) is strongly clean for all n 1 and that the group ring of a locally finite group over a strongly regular or commutative strongly π-regular ring is strongly clean.

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.

Rings in which certain right ideals are direct summands of annihilators

This paper is a continuation of the study of the rings for which every principal right ideal (respectively, every right ideal) is a direct summand of a right annihilator initiated by Stanley S. Page and the author in [20, 21].

The lattice of pre-natural classes of modules☆

Abstract In this paper, we introduce and study a new complete lattice whose elements are the so-called pre-natural classes of R-modules. This lattice contains a complete sublattice isomorphic to the complete lattice of all linear topologies of R and a sublattice anti-isomorphic to the frame of all hereditary torsion theories of R. Moreover, the complete Boolean lattice of all natural classes of R-modules is a sublattice of this lattice. Various properties of this new lattice are formulated and some applications to the ring R are given.