Armendariz group rings

For a torsion or torsion-free group G and a field F, we characterize the group algebra FG that is Armendariz. Armendariz property for a group ring over a general ring R is also studied and related to those of Abelian group rings and the quaternion ring over R.     

On generalized stable rings

In this paper, we introduce the class of generalized stable rings and investigate equivalent characterizations of such rings. We show that End_R R is a generalized stable ring if and only if any right H-module decompositions M = A ₁ + B ₁ = A ₂ + B ₂ with A ₁ ? A ? A ₂ implies that there exist C,D,E ≤ M such that M = C + D B ₁ = c + E + B ₂ with.C ? A. Also we show that every generalized stable ring is a GE-ring and matrices over generalized stable regular ring can be diagonalized by some weakly invertible matrices.     

Strong separativity over exchange rings

An exchange ring R is strongly separative provided that for all finitely generated projective right R-modules A and B, A ⊕ A ≅ A ⊕ B ⇒ A ≅ B. We prove that an exchange ring R is strongly separative if and only if for any corner S of R, aS + bS = S implies that there exist u, v ∈ S such that au = bv and Su + Sv = S if and only if for any corner S of R, aS + bS = S implies that there exists a right invertible matrix ∈ M ₂(S). The dual assertions are also proved.     

Commutative neat group rings

A ring R is called clean if every element of it is a sum of an idempotent and a unit. A ring R is neat if every proper homomorphic image of R is clean. When R is a field, then a complete characterization has been obtained for a commutative group ring RG to be neat, but not clean. And if R is not a field, then necessary conditions are obtained for a commutative group ring RG to be neat, but not clean. A counterexample is given to show that these necessary conditions are not sufficient.     

Weakly left localizable rings

A new class of rings, the class of weakly left localizable rings, is introduced. A ring R is called weakly left localizable if each non-nilpotent element of R is invertible in some left localization S^?1R of the ring R. Explicit criteria are given for a ring to be a weakly left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case for all left Noetherian rings). It is proved that a ring with finitely many maximal left denominator sets that satisfies some natural conditions is a weakly left localizable ring iff its left quotient ring is a direct product of finitely many local rings such that their radicals are nil ideals.     

A Note on Semilocal Group Rings

Let R be an associative ring with identity and let J(R) denote the Jacobson radical of R. R is said to be semilocal if R/J(R) is Artinian. In this paper we give necessary and sufficient conditions for the group ring RG, where G is an abelian group, to be semilocal.     

Some intersections and identifications in integral group rings

LetZG be the integral group ring of a groupG and I(G) its augmentation ideal. For a free groupF andR a normal subgroup ofF, the intersectionI ⁿ⁺¹ (F) ∩I ⁿ⁺¹ (R) is determined for alln≥ 1. The subgroupsF ∩ (1+ZFI (R) I (F) I (S)) ANDF ∩ (1 + I (R)I ³ (F)) of F are identified whenR and S are arbitrary subgroups ofF.     

On rings with associated elements

A principal right ideal of a ring is called uniquely generated if any two elements of the ring that generate the same principal right ideal must be right associated (i.e., if for all a,b in a ring R, aR = bR implies a = bu for some unit u of R). In the present paper, we study “uniquely generated modules” as a module theoretic version of “uniquely generated ideals,” and we obtain a characterization of a unit-regular endomorphism ring of a module in terms of certain uniquely generated submodules of the module among some other results: End(M) is unit-regular if and only if End(M) is regular and all M-cyclic submodules of a right R-module M are uniquely generated. We also consider the questions of when an arbitrary element of a ring is associated to an element with a certain property. For example, we consider this question for the ring R[x;σ]∕(xⁿ⁺¹), where R is a strongly regular ring with an endomorphism σ be an endomorphism of R.     

A note on utumi quotient rings

Let R be a ring and ρ a right ideal of R with zero right annihilate. Then ρ and R have the same left Utumi quotient ring. We study the lifting properties of GPIs and some chain conditions inherited by such right ideals. Next, we prove a generalization of Chatters’ theorem. Precisely, we show that if R is a right nonsingular ring with finite right Goldie dimension and possesses a right ideal ρ such that both ρ and l _R(ρ) are PI-rings, then the right Utumi quotient ring of R is also a Pi-ring.     

On inverse skew Laurent series extensions of weakly rigid rings

Let R be a ring equipped with an automorphism α and an α-derivation δ. We studied on the relationship between the quasi Baerness and (α, δ)-quasi Baerness of a ring R and these of the inverse skew Laurent series ring R((x^?1; α, δ)), in case R is an (α, δ)-weakly rigid ring. Also we proved that for a semicommutative (α, δ)-weakly rigid ring R, R is Baer if and only if so is R((x^?1; α, δ)). Moreover for an (α, δ)-weakly rigid ring R, it is shown that the inverse skew Laurent series ring R((x^?1; α, δ)) is left p.q.-Baer if and only if R is left p.q.-Baer and every countable subset of left semicentral idempotents of R has a generalized countable join in R.     

Homomorphisms between A-Projective Abelian Groups and Left Kasch-Rings

Glaz and Wickless introduced the class G of mixed abelian groups A which have finite torsion-free rank and satisfy the following three properties: i) A _p is finite for all primes p, ii) A is isomorphic to a pure subgroup of _P A _P and iii) Hom(A, tA) is torsion. A ring R is a left Kasch ring if every proper right ideal of R has a non-zero left annihilator. We characterize the elements A of G such that E(A)/tE(A) is a left Kasch ring, and discuss related results.     

Uniserial Noetherian centrally essential rings

Abstract

It is proved that a ring R is a right uniserial, right Noetherian centrally essential ring if and only if R is a commutative discrete valuation domain or a left and right Artinian, left and right uniserial ring. It is also proved that there exist non-commutative uniserial Artinian centrally essential rings.     

Projective ideals of skew polynomial rings over HNP rings

Let R be an hereditary Noetherian prime ring (or, HNP-ring, for short), and let S?=?R[x;σ] be a skew polynomial ring over R with σ being an automorphism on R. The aim of the paper is to describe completely the structure of right projective ideals of R[x;σ] where R is an HNP-ring and to obtain that any right projective ideal of S is of the form X𝔟[x;σ], where X is an invertible ideal of S and 𝔟 is a σ-invariant eventually idempotent ideal of R.     

Matrices over Zhou nil-clean rings

A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Let R be a Zhou nil-clean ring. If R is 2-primal (of bounded index), we prove that every square matrix over R is the sum of two tripotents and a nilpotent. These provide a large class of rings over which every square matrix has such decompositions by tripotent and nilpotent matrices.     

Semiprimeness,quasi-Baerness and prime radical of skew generalized power series rings

Let R be a ring, (S,≤) a strictly ordered monoid and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev-Neumann Laurent series rings. In this paper we obtain necessary and sufficient conditions for the skew generalized power series ring R[[S,ω]] to be a semiprime, prime, quasi-Baer, or Baer ring. Furthermore, we study the prime radical of a skew generalized power series ring R[[S,ω]]. Our results extend and unify many existing results. In particular, we obtain new theorems on (skew) group rings, Mal’cev-Neumann Laurent series rings and the ring of generalized power series.     

Clean elements in abelian rings

Let R be a ring with identity. An element in R is said to be clean if it is the sum of a unit and an idempotent. R is said to be clean if all of its elements are clean. If every idempotent in R is central, then R is said to be abelian. In this paper we obtain some conditions equivalent to being clean in an abelian ring.     

Elementary matrix reduction over certain rings

We explore elementary matrix reduction over certain rings characterized by properties related to stable range. Let R be a commutative ring. We call R locally stable if aR+bR = R??x∈R such that R∕(a+bx)R has stable range 1. We study locally stable rings and prove that every locally stable Bézout ring is an elementary divisor ring. Many known results on domains are thereby generalized.     

Principal quasi-Baerness of formal power series rings

Let R be a ring. We consider left （or right） principal quasi-Baerness of the left skew formal power series ring R[[x;α]] over R where a is a ring automorphism of R. We give a necessary and sufficient condition under which the ring R[[x; α]] is left （or right） principally quasi-Baer. As an application we show that R[[x]] is left principally quasi-Baer if and only if R is left principally quasi- Baer and the left annihilator of the left ideal generated by any countable family of idempotents in R is generated by an idempotent.     

Homomorphic images of subdirectly irreducible rings

We prove that every ring is a proper homomorphic image of some subdirectly irreducible ring. We also show that a finite ring R does not need to be isomorphic to the factor of a subdirectly irreducible ring by its monolith as well as R does not need to be a homomorphic image of a finite subdirectly irreducible ring. We provide an analogous characterization also for varieties of rings with unity, for the quasiregular rings, for the rings with involution and for their subvarieties of commutative rings.