Reversible Group Rings Over Commutative Rings

Let G be a torsion group and R be a commutative ring with identity. We investigate reversible group rings RG over commutative rings, extending results of Gutan and Kisielewicz which characterize all reversible group rings over fields.     

Maximal divisible subgroups in modular group rings of p-mixed abelian groups

The isomorphism structure of the maximal divisible subgroup of the subgroup V _p (R(G); H) Id R(G) of the normalized unit group V R(G) in a commutative group ring R(G) is completely described only in terms of R, G and H whenever R is a commutative unital ring of prime characteristic p and G is a p-mixed abelian group. In particular, the maximal divisible subgroup of V R(G) is characterized. This extends a result due to Nachev (Commun. Algebra, 1995) as well as a result due to the author (Commun. Algebra, 2010).     

Weakly Stable Modules

ABSTRACT

A new notion which is called weakly stable module is introduced in this article. It is a nontrivial generalization of the modules with endomorphism rings having stable range one. We deduce that weakly stable projective modules have the cancellation property, and so any commutative hereditary ring has the cancellation property, i.e., if R is a commutative hereditary ring, then for any R-modules B and C, R ⊕ B ? R ⊕ C implies B ? C.     

Reciprocal Polynomials and p-Group Cohomology

Let p be a prime and let G be a finite p-group. In a recent paper (Woodcock, J Pure Appl Algebra 210:193–199, 2007) we introduced a commutative graded ?-algebra R _G . This classifies, for each commutative ring R with identity element, the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element. We show here that, up to inseparable isogeny, the “graded-commutative” mod p cohomology ring $H^\ast(G, \mathbb{F}_p)Let p be a prime and let G be a finite p-group. In a recent paper (Woodcock, J Pure Appl Algebra 210:193–199, 2007) we introduced a commutative graded ℤ-algebra R _G . This classifies, for each commutative ring R with identity element, the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element. We show here that, up to inseparable isogeny, the “graded-commutative” mod p cohomology ring H^*(G, \mathbbF_p)H^\ast(G, \mathbb{F}_p) of G has the same spectrum as the ring of invariants of R _G mod p (R_G ?_\mathbbZ \mathbbF_p)^G(R_G \otimes_{\mathbb{Z}} \mathbb{F}_p)^G where the action of G is induced by conjugation.     

On maximal commutative subrings of non-commutative rings

We observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.     

On weakly nil-clean rings

We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of an abelian weakly nil-clean ring, and of a strongly nil-clean ring. As applications, this result is used to determine the 2-primal rings R such that the matrix ring \(\mathbb{M}_n (R)\) is weakly nil-clean, and to show that the endomorphism ring End _D (V) over a vector space V _D is weakly nil-clean if and only if it is nil-clean or dim(V) = 1 with D?= ?₃.     

Semiclean Rings

Abstract

The notion of semiclean elements in a ring is defined. Every clean element is semiclean. A ring R is said to be semiclean if every element in R is semiclean. The group ring Z _p G with G a cyclic group of order 3 is proved to be semiclean. The n × n matrix ring M _n (R) over a semiclean ring is semiclean. If R is a torsion free semiclean ring in which every element of R can be written as a sum of periodic and ±1, then R is clean. Every element in a semiclean ring R with 2 invertible is a sum of no more than 3 units.     

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.     

Orders in regular rings with minimal condition for principal right ideals

ABSTRACT

A ring R is called an n-clean (resp. Σ-clean) ring if every element in R is n-clean (resp. Σ-clean). Clean rings are 1-clean and hence are Σ-clean. An example shows that there exists a 2-clean ring that is not clean. This shows that Σ-clean rings are a proper generalization of clean rings. The group ring ?_(p) G with G a cyclic group of order 3 is proved to be Σ-clean. The m× m matrix ring M _m (R) over an n-clean ring is n-clean, and the m×m (m>1) matrix ring M _m (R) over any ring is Σ-clean. Additionally, rings satisfying a weakly unit 1-stable range were introduced. Rings satisfying weakly unit 1-stable range are left-right symmetric and are generalizations of abelian π-regular rings, abelian clean rings, and rings satisfying unit 1-stable range. A ring R satisfies a weakly unit 1-stable range if and only if whenever a ₁ R + ˙˙˙ a _m R = dR, with m ≥ 2, a ₁,…, a _m, d ∈ R, there exist u ₁ ∈ U(R) and u ₂,…, u _m ∈ W(R) such that a ₁ u ₁ + ? a _m u _m = Rd.     

On weakly prime radical of modules and semi-compatible modules

Summary Let M be a left R-module. Then a proper submodule P of M is called weakly prime submodule if for any ideals A and B of R and any submodule N of M such that ABN ⊆ P, we have AN ⊆ P or BN ⊆ P. We define weakly prime radicals of modules and show that for Ore domains, the study of weakly prime radicals of general modules reduces to that of torsion modules. We determine the weakly prime radical of any module over a commutative domain R with dim (R) ≦ 1. Also, we show that over a commutative domain R with dim (R) ≦ 1, every semiprime submodule of any module is an intersection of weakly prime submodules. Localization of a module over a commutative ring preserves the weakly prime property. An R-module M is called semi-compatible if every weakly prime submodule of M is an intersection of prime submodules. Also, a ring R is called semi-compatible if every R-module is semi-compatible. It is shown that any projective module over a commutative ring is semi-compatible and that a commutative Noetherian ring R is semi-compatible if and only if for every prime ideal B of R, the ring R/\B is a Dedekind domain. Finally, we show that if R is a UFD such that the free R-module R⊕ R is a semi-compatible module, then R is a Bezout domain.     

Mod-Retractable Rings

A right module M over a ring R is said to be retractable if Hom _R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ? _R RG is a retractable RG-module if and only if M _R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.     

Homological Characterizations of Rings with Property (P)

Abstract

A commutative ring R is said to satisfy property (P) if every finitely generated proper ideal of R admits a non-zero annihilator. In this paper we give some necessary and sufficient conditions that a ring satisfies property (P). In particular, we characterize coherent rings, noetherian rings and Π-coherent rings with property (P).     

Dubrovin valuation properties of skew group rings and crossed products

In this paper some conditions for a skew group ring or a crossed product to have finite weak global dimension are given.Using these results we obtain some necessary conditions and some sufficient conditions for a skew group ring or a crossed product to be a Dubrovin valuation ring.If R*G is a skew group ring, where the coefficient ring R is a commutative ring and G is a finite group, then we prove that the conditions we obtained become necessary and sufficient conditions.In particular, if R is a commutative valuation ring, then R*G is a Dubrovin valuation ring if and only if G _T=<1>,where G _T is the inertial group of R.     

Polynomials Contained in a Finite Number of Maximal Ideals

Abstract

Let R[X] be a polynomial ring in one variable over a commutative ring R. If (R,?) is a local ring then any Weierstrass polynomial in R[X] is contained only in the maximal ideal (?,X) of R[X]. We generalise this property of Weierstrass polynomials and investigate properties of polynomials contained in a finite number of maximal ideals in R[X].     

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.     

On the Commutative Twisted Group Algebras

Let G be an abelian group and let R be a commutative ring with identity. Denote by R _t G a commutative twisted group algebra (a commutative twisted group ring) of G over R, by ?(R) and ?(R _t G) the nil radicals of R and R _t G, respectively, by G _p the p-component of G and by G ₀ the torsion subgroup of G. We prove that:

1. If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and ?(R) = 0, then there exists a twisted group algebra R _{t 1} (G/G _p ) such that R _t G/?(R _t G) ? R _{t 1} (G/G _p ) as R-algebras;

2. If R is a ring of prime characterisitic p and R* is p-divisible, then ?(R _t G) = 0 if and only if ?(R) = 0 and G _p  = 1; and

3. If B(R) = 0, the orders of the elements of G ₀ are not zero divisors in R, H is any group and the commutative twisted group algebra R _t G is isomorphic as R-algebra to some twisted group algebra R _{t 1} H, then R _t G ₀ ? R _{t 1} H ₀ as R-algebras.

     

The Converse of Schur's Lemma in Noetherian Rings and Group Algebras

ABSTRACT

If M is a simple module over a ring R, then, by Schur's Lemma, its endomorphism ring is a division ring. However, the converse of this property, which we called the CSL property, does not hold in general. The object of this article is to study this converse for a few classes of rings: left Noetherian rings, V-rings and group algebras. First, we establish that a left Noetherian ring R is a CSL ring if and only if a ring R is left–artinian and primary decomposable. Secondly, we prove that a left semiartinian V-ring is CSL. At last, we study the CSL property in group algebra K [ G ] where K a field algebraically closed of characteristic p and G is a finite group of order divisible by p. Our main contribution is that K [ G ] is a CSL ring if and only if Gbf = HP where H is a normal p′-subgroup and bfP a Sylow bfp-subgroup of bfG. In this case, K [ G ] is primary decomposable.     

Skew Group Rings which are Semi-Hereditary Orders and Prüfer Orders in Simple Artinian Rings

Let R be a commutative ring, let G be a finite group acting on R as automorphisms of R and let R * G be the skew group ring. By using the decomposition subgroups of G, the inertial subgroups of G, the properties of the coefficient ring R and the properties of the fixed subring R ^G , some necessary and sufficient conditions for R * G to be a prime Goldie ring, a semi-hereditary order in a simple Artinian ring, or a Prüfer order in a simple Artinian ring are given.     

Commutative rings whose proper ideals are direct sums of uniform modules

An interesting result, obtaining by some theorems of Asano, Köthe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ? of the form ? = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (resp., cocyclic modules). Examples which delineate the structures are provided.