Rings and covered groups (original) (raw)

A Study on Algebraic Structures of Groups and Rings

Indian journal of natural sciences, 2023

The aim of this paper is to present the basic concepts of abstract algebra like groups, rings, integral domain (ID),) & also to study about the properties of these concepts. But the main focus is to study about the relationship, generalization of these concepts. Throughout this paper we considers ring it may be both commutative as well as non-commutative as the case may be & group it may be abelian or nonabelian as required.

On group rings: a translation of "Sur les anneaux de groupes" by Guy Renault

2018

We characterize the rings A and groups G for which the group rings A[G] are local, semi-local, or left perfect [14]. The recent work of M. P. Malliavin [13] and J. L. Pascaud permits the completion of results of [14] on self-injective group rings. A designates a ring with identity but which is not necessarily commutative, and G is a group. The fields involved are not necessarily commutative. For an exposition on group rings, consult J. Lambek [12] and P. Ribenboim [15].

3 RINGS ASSOCIATED TO COVERINGS OF FINITE p-GROUPS

2016

There is a natural way to associate a ring to any group. In this paper we characterize the rings associated to finite p-groups when the covering consists of maximal cyclic subgroups and subgroups of order p 2. 2010 Mathematics Subject Classification. 16S60. Key words and phrases. finite p-groups, covers of groups, rings of functions. Research of the second author supported by the Louisiana BoR [LEQSF(2012-15)-RD-A-20].

Rings associated to coverings of finite p-groups

There is a natural way to associate a ring to any group. In this paper we characterize the rings associated to finite p-groups when the covering consists of maximal cyclic subgroups and subgroups of order p^2.

Extending abelian groups to rings

Journal of the Australian Mathematical Society, 2007

For any abelian group G and any function f : G → G we define a commutative binary operation or "multiplication" on G in terms of f . We give necessary and sufficient conditions on f for G to extend to a commutative ring with the new multiplication. In the case where G is an elementary abelian p-group of odd order, we classify those functions which extend G to a ring and show, under an equivalence relation we call weak isomorphism, that there are precisely six distinct classes of rings constructed using this method with additive group the elementary abelian p-group of odd order p 2 .

Notes on abelian groups. II

Acta Mathematica Academiae Scientiarum Hungaricae

w 5. p.basic subgroups of arbitrary abelian groups KULIKOV [8] introduced the notion of basic subgroups of abelian p-groups which has proved to be one of the most important notions in the theory of p-groups of arbitrary power. Basic subgroups can be defined in any module over the ring of p-adic integers, or, more generally, over any discrete valuation ring. Here we want to give a generalization of basic subgroups to any group so that it coincides with the old concept whenever the group is primary. In the general case, to every prime p, one can define p-basic subgroups where in the definition the prime p plays a distinguished role. The p-basic subgroups are not isomorphic for different primes, but are uniquely determined (up to isomorphism) by the group and the prime p. We shall see that p-basic subgroups are useful in certain investigations. Let G be an arbitrary (abelian) group l and p an arbitrary, but fixed prime. We call a subset [x~]~ea of G, not containing 0, p-independent, if for any finite subset xl .... ,x~ a relation nlxl-[-... q-nkx1~ EprG

On Abelian Rings

2010

Let α be an endomorphism of an arbitrary ring R with identity. In this note, we introduce the notion of α-abelian rings which generalizes abelian rings. We prove that α-reduced rings, α-symmetric rings, α-semicommutative rings and α-Armendariz rings are α-abelian. For a right principally projective ring R , we also prove that R is α-reduced if and only if R is α-symmetric if and only if R is α-semicommutative if and only if R is α-Armendariz if and only if R is α-Armendariz of power series type if and only if R is α-abelian. Key word and phrases: α-reduced rings, α-symmetric rings, α-semicommutative rings, α-Armendariz rings, α-abelian rings.

3 Abstract Homomorphisms of Algebraic Groups and Applications

2016

We give a survey of recent rigidity results for linear representations of elementary subgroups of Chevalley groups over commutative rings, and of other similar groups, and also discuss some applications to deformations of representations of elementary groups over finitely generated commutative rings.