广义半π-正则环

In this paper, the concept of right generalized semi-w-regular rings is defined. We prove that these rings are non-trival generalizations of both right GP-injective rings and semi- π-regular rings. Some properties of these rings are studied and some results about generalized semiregular rings and GP-injective rings are extended.     

Group homomorphisms inducing cohomology monomorphisms

Let be a homomorphism of -groups such that
is injective, for . We prove that the non-bijectivity of implies the existence of a quotient of containing as a proper direct factor. This gives a refined proof of a result of Evens, which asserts that is bijective if is.

     

I半π正则环

Let R be a ring and I an ideal of R.A ring R is called I-semi-π-regular if R/I isπ-regular and idempotents of R can be strongly lifted modulo I.Charac- terizations of I-semi-π-regular rings are given and relations between semi-π-regular rings and semiregular rings are explored.     

关于Jacobson根的一些研究

主要证明了:(i)假设R是右广义半正则右ACS-环,若J(R)∩I=J(I)对于R的任意右理想I都成立,则J(R)=Z(RR);(ii)如果R是右AP-内射环且R的每个奇异单右R-模是GP-内射,则对于R的任意右理想I都有J(R)∩I=J(I).     

Generalized Burnside rings and group cohomology

In this paper we give by a unified formula the classification of exceptional compact simple Kantor triple systems defined on tensor products of composition algebras corresponding to realifications of exceptional simple Lie algebras.     

Group Identities and Prime Rings Generated by Units

Abstract

Let R be a prime ring and U(R) the group of units of R. We prove that if U(R) generates R and satisfies a group identity,then R is either a domain or a full matrix ring over a finite field.     

Study of Hopficity in Certain Classes of Rings

The main results proved in this paper are:

1. For any non-zero vector space V _Dover a division ring D, the ring R= End(V _D) is hopfian as a ring

2. Let Rbe a reduced π-regular ring &; B(R) the boolean ring of idempotents of R. If B(R) is hopfian so is R.The converse is not true even when Ris strongly regular.

3. Let Xbe a completely regular spaceC(X) (resp. C ^?(X)) the ring of real valued (resp. bounded real valued) continuous functions on X. Let Rbe any one of C(X) or C ^?(X). Then Ris an exchange ring if &; only if Xis zero dimensional in the sense of Katetov. for any infinite compact totally disconnected space X C(X) is an exchange ring which is not von Neumann regular.

4. Let Rbe a reduced commutative exchange ring. If Ris hopfian so is the polynomial ring R[T ₁,…,T _n] in ncommuting indeterminates over Rwhere nis any integer ≥ 1.

5. Let Rbe a reduced exchange ring. If Ris hopfian so is the polynomial ring R[T].     

Generalized Derivations and nth Power Maps in Rings

We prove that a Jordan (φ, θ)-derivation of a 2-torsion free semiprime ring, with θ an automorphism, must be a (φ, θ)-derivation. This is then used to show more generally that an additive mapping of a semiprime ring to itself that behaves like a generalized (φ, θ)-derivation on nth powers is, in fact, a generalized (φ, θ)-derivation of the ring.     

Generalized Ulam-Hyers stability of random homomorphisms in random normed algebras associated with the Cauchy functional equation

Using the fixed point method, we prove the generalized Ulam-Hyers stability of random homomorphisms in random normed algebras associated with the Cauchy functional equation.     

Central Units in Metacyclic Integral Group Rings

In this article, we give a method to compute the rank of the subgroup of central units of ? G, for a finite metacyclic group, G, by means of ?-classes and ?-classes. Then we construct a multiplicatively independent set  ? (U(? C _{p, q} )) and by applying our results, we prove that  generates a subgroup of finite index.     

Torsion Units in the Integral Group Ring of the Alternating Group of Degree 6

It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring ?G of a finite group G conjugates to a group element within the rational group algebra ?G. We investigate the Zassenhaus Conjecture (ZC) and a conjecture by W. Kimmerle about prime graph in the normalized unit group of ?A₆.     

Unique factorization in generalized power series rings

Let be a field of characteristic zero and let denote the ring of generalized power series (i.e., formal sums with well-ordered support) with coefficients in , and non-positive real exponents. Berarducci (2000) constructed an irreducible omnific integer, in the sense of Conway (2001), by first proving that an element of that is not divisible by a monomial and whose support has order type (or for some ordinal ) must be irreducible. In this paper, we consider elements of with support of order type . The irreducibility of these elements cannot be deduced solely from the order type of their support and, after developing new tools for studying these elements, we exhibit both reducible and irreducible elements of this type. We further prove that all elements whose support has order type and which are not divisible by a monomial factor uniquely into irreducibles. This provides, in the ring , a class of reducible elements for which we have unique factorization into irreducibles.

     

Maximal Divisible Subgroups in p-Mixed Modular Abelian Group Rings

The isomorphism structure of the maximal divisible subgroup dVR(G) of the normed unit group VR(G) in a commutative group ring R(G) is completely described only in terms of R and G whenever R is a commutative unital ring of prime characteristic p and G is a p-mixed abelian group. This somewhat extends a result due to Nachev [13 Nachev , N. A. ( 1995 ). Invariants of the Sylow p-subgroup of the unit group of a commutative group ring of characteristic p. Commun. Algebra 23 : 2469 – 2489 . [Google Scholar]] as well as results due to the author [1 Danchev , P. V. ( 2004 ). Maximal divisible subgroups in modular group algebras of p-mixed and p-splitting abelian groups . Rad. Mat. 13 : 23 – 32 . [Google Scholar]] and Kuneva [11 Kuneva , V. ( 2006 ). Maximal divisible subgroups in modular group algebras of p-mixed abelian groups . Compt. rend. Acad. bulg. Sci. 59 : 899 – 902 . [Google Scholar]].     

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.     

Factorization in generalized power series

The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): . Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway's series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is either or of the form and the series is not divisible by any monomial, then it is irreducible. To handle the general case we use a suggestion of
M.-H. Mourgues, based on an idea of Gonshor, which allows us to reduce to the special case . In the final part of the paper we study the irreducibility of series with finite support.

     

Generalized Blocks of Unipotent Characters in the Finite General Linear Group

In an article of 2003, Külshammer, Olsson, and Robinson defined ?-blocks for the symmetric groups, where ? > 1 is an arbitrary integer, and proved that they satisfy an analogue of the Nakayama Conjecture. Inspired by this work and the definitions of generalized blocks and sections given by the authors, we give in this article a definition of d-sections in the finite general linear group, and construct d-blocks of unipotent characters, where d ≥ 1 is an arbitrary integer. We prove that they satisfy one direction of an analogue of the Nakayama Conjecture, and, in some cases, prove the other direction. We also prove that they satisfy an analogue of Brauer's Second Main Theorem.     

A Delocalization Property for Assembly Maps and an Application in Algebraic K-Theory of Group Rings

Let be a group, F the free -module on the set of finite order elements in , with acting by conjugation, and the ring extension of by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaWaaiWaaeaada% WcaaqaaiaaigdaaeaatCvAUfKttLearyGqLXgBG0evaGqbciab-5ga% UbaaieaacaGFLbGaaGOmaiaabc8acqWFPbqAcaqGVaGae8NBa42aaq% qaaeaacqGHdicjcqaHZoWzcqGHiiIZcqqHtoWrcaqGGaGaae4Baiaa% bAgacaqGGaGaae4BaiaabkhacaqGKbGaaeyzaiaabkhacaqGGaGae8% NBa4gacaGLhWoaaiaawUhacaGL9baaaaa!563E!\[\left\{ {\frac{1}{n}e2{\text{\pi }}i{\text{/}}n\left| {\exists \gamma \in \Gamma {\text{ of order }}n} \right.} \right\}\]. For a ring R with , we build an injective assembly map , detected by the Dennis trace map. This is proved by establishing a delocalization property for the assembly map in Hochschild homology, namely providing a gluing of simpler assembly maps (i.e. localized at the identity of ) to build , and by delocalizing a known assembly map in K-theory to define . We also prove the delocalization property in cyclic homology and in related theories.