Group Ring RG Research Articles

Annihilator graphs derived from group rings

Let RG be the group ring of the group G over a ring R and let Z∗(RG) be the collection of all non-zero zero divisors in a finite group ring RG. And, for x ∈ Z∗(RG), ann(x) = { r ∈ RG | rx = 0}. The annihilator graph of RG denoted as AG(RG), and is defined as the graph whose vertex set is the elements of non-zero zero divisors in RG and two distinct vertices vx and vy are adjacent if and only if ann(x) ∩ ann(y) ≠ 0. In this paper we try to characterize the properties of annihilator graphs in group rings.

Rings close to periodic with applications to matrix, endomorphism and group rings

We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl’s question whether or not a ring R being nil-clean implies that the matrix ring M n ( R ) over R is also nil-clean for all n ≥ 1 is paralleling to the corresponding implication for (abelian, local) periodic rings. Besides, we study when the endomorphism ring E ( G ) of an abelian group G is periodic. Concretely, we establish that E ( G ) is periodic exactly when G is finite as well as we find a complete necessary and sufficient condition when the endomorphism ring over an abelian group is strongly m-nil clean for some natural number m thus refining an “old” result concerning strongly nil-clean endomorphism rings. Responding to a question when a group ring is periodic, we show that if R is a right (resp., left) perfect periodic ring and G is a locally finite group, then the group ring RG is periodic, too. We finally find some criteria under certain conditions when the tensor product of two periodic algebras over a commutative ring is again periodic. In addition, some other sorts of rings very close to periodic rings, namely the so-called weakly periodic rings, are also investigated.

On Modules over G-sets

Let R be a commutative ring with unity, M a module over R and let S be a G–set for a finite group G. We define a set MS to be the set of elements expressed as the formal finite sum of the form ∑s∈Smss where ms∈M. The set MS is a module over the group ring RG under the addition and the scalar multiplication similar to the RG–module MG. With this notion, we not only generalize but also unify the theories of both of the group algebra and the group module, and we also establish some significant properties of (MS)RG. In particular, we describe a method for decomposing a given RG–module MS as a direct sum of RG–submodules. Furthermore, we prove the semisimplicity problem of (MS)RG with regard to the properties of MR, S and G.

Değişmeli Grup Halkalarında G-Nilpotent Birimsel Elemanların Direkt Çarpım Gruplarına Bir Genellemesi

Let V(RG) denote the normalized unit group of the group ring RG of a group G over a ring R. The concept of G-nilpotent unit in a commutative group ring has been defined in (Danchev, 2012). In this study, some necessary and sufficient conditions for a normalized unit group in a commutative group ring of a direct product group G×H to consist only of G×H-nilpotent units have been given and especially some results which are related to groups G×C_3 and G×C_4 have been introduced where C_3 and C_4 are cyclic groups of orders 3 and 4 respectively. In this context, we can say that the paper extends the results in (Danchev, 2012). At the end, an open problem is served as a future work.

On the probability of zero divisor elements in group rings

Let R be a non trivial finite commutative ring with identity and G be a non trivial group. We denote by P(RG) the probability that the product of two randomly chosen elements of a finite group ring RG is zero. We show that P(RG) <0.25 if and only if RG is not isomorphic to Z2C2, Z3C2, Z2C3. Furthermore, we give the upper bound and lower bound for P(RG). In particular, we present the general formula for P(RG), where R is a finite field of characteristic p and |G| ≤ 4.

The twisted derivation problem for group rings

We study $$(\sigma ,\tau )$$ -derivations of a group ring RG where G is a group with center having finite index in G and R is a semiprime ring with 1 such that either R has no torsion elements or that if R has p-torsion elements, then p does not divide the order of G and let $$\sigma ,\tau $$ be R-linear endomorphisms of RG fixing the center of RG pointwise. We generalize Main Theorem 1.1 of Chaudhuri (Comm Algebra 47(9): 3800–3807, 2019) and prove that there is a ring $$T\supset R$$ such that $$\mathcal {Z}(T)\supset \mathcal {Z}(R)$$ and that for the natural extensions of $$\sigma , \tau $$ to TG, we get $$H^1(TG,{}_\sigma TG_\tau )=0$$ , where $${}_\sigma TG_\tau $$ is the twisted $$TG-TG$$ -bimodule. We provide applications of the above result and Main Theorem 1.1 of Chaudhuri (2019) to integral group rings of finite groups and connect twisted derivations of integral group rings to other important problems in the field such as the isomorphism problem and the Zassenhaus conjectures. We also give an example of a group G which is both locally finite and nilpotent and such that for every field F, there exists an F-linear $$\sigma $$ -derivation of FG which is not $$\sigma $$ -inner.

A cancellation theorem for modules over integral group rings

AbstractA long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups G for which the integral group ring ℤG has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that ℤG has SFC provided at most one copy of the quaternions ℍ occurs in the Wedderburn decomposition of the real group ring ℝG. This generalises the Eichler condition in the case of integral group rings.

On Idempotent Units in Commutative Group Rings

Special elements as units, which are defined utilizing idempotent elements, have a very crucial place in a commutative group ring. As a remark, we note that an element is said to be idempotent if r^2=r in a ring. For a group ring RG, idempotent units are defined as finite linear combinations of elements of G over the idempotent elements in R or formally, idempotent units can be stated as of the form id(RG)={∑_(r_g∈id(R))▒〖r_g g〗: ∑_(r_g∈id(R))▒r_g =1 and r_g r_h=0 when g≠h} where id(R) is the set of all idempotent elements [3], [4], [5], [6]. Danchev [3] introduced some necessary and sufficient conditions for all the normalized units are to be idempotent units for groups of orders 2 and 3. In this study, by considering some restrictions, we investigate necessary and sufficient conditions for equalities:i.V(R(G×H))=id(R(G×H)),ii.V(R(G×H))=G×id(RH),iii.V(R(G×H))=id(RG)×Hwhere G×H is the direct product of groups G and H. Therefore, the study can be seen as a generalization of [3], [4]. Notations mostly follow [12], [13].

English

A ring R is called right P-injective if every homomorphism from a principal right ideal of R to RR can be extended to a homomorphism from RR to RR. Let R be a ring and G a group. Based on a result of Nicholson and Yousif, we prove that the group ring RG is right P-injective if and only if (a) R is right P-injective; (b) G is locally finite; and (c) for any finite subgroup H of G and any principal right ideal I of RH, if f ∈ HomR(IR,RR), then there exists g ∈ HomR(RHR,RR) such that gI = f. Similarly, we also obtain equivalent characterizations of n-injective group rings and F-injective group rings.

Anticommutativity of symmetric elements under generalized oriented involutions

Let R be a ring with whose unit group is denoted by G a group with involution and a nontrivial group homomorphism, with satisfying for all Let RG be the group ring of G over R and define the involution in RG by In this article, we will classify the group rings RG such that is anticommutative, where is the largest subset of that can satisfy anticommutativity under Besides that, previous results about or will be extended.Communicated by S. Sehgal

Commutative weakly nil-neat group rings

Let R be a ring and let G be a group. We prove a rather curious necessary and sufficient condition for the commutative group ring RG to be weakly nil-neat only in terms of R,G and their sections. This somewhat expands three recent results, namely those established by McGovern et al. in (J. Algebra Appl., 2015), by Danchev-McGovern in (J. Algebra, 2015) and by the present authors in (J. Math., Tokushima Univ., 2019), related to commutative nil-clean, weakly nil-clean and nil-neat group rings, respectively.

A new characterization of commutative semiregular rings

A commutative ring R is called J-rad clean in case, for any r ∈ R, there is an idempotent e ∈ R such that r−e ∈ U(R) and re ∈ J(R), where U(R) and J(R) denote the set of units and the Jacobson radical of R, respectively. Also, a ring R is called semiregular if R/J(R) is regular in the sense of von Neumann and idempotents lift modulo J(R). We demonstrate that these two concepts are, actually, equivalent and we portray a portion of the properties of this class of rings. In particular, as a direct application, we prove that the commutative group ring RG is J-rad clean if, and only if, R is a commutative J-rad clean ring and G is a torsion abelian group, provided that J(R) is nil.

Lie centrally metabelian group rings over noncommutative rings

Let R be a unital ring and let G be an arbitrary nonabelian group. Lie centrally metabelian group algebras have been studied and described by various authors. In this article, we have classified Lie centrally metabelian group rings over noncommutative rings. We also give characterizations of group rings RG satisfying the Lie identity over both commutative and noncommutative rings.

Weakly Lie Engel condition on group rings

Let R be a commutative ring of characteristic and G be a group. It is known that the group ring RG is bounded Lie Engel if and only if either G is nilpotent and G has a p-Abelian normal subgroup of finite p-power index (if n is a power of a prime p) or G is Abelian. In this paper we try to generalize this result; if x and y are elements of RG, let [x,y] = xy – yx and inductively, . Let m and n be two natural numbers. Among other results, we show that if RG satisfies , then either is a p-group (if n is a power of a prime p) or G is Abelian. If G is locally finite, then we show that RG satisfies if and only if either G is locally nilpotent and is a p-group (if n is a power of a prime p) or G is Abelian.

-Derivations of group rings

We study -derivations of a group ring RG of a finite group G over an integral domain R with 1. As an application we extend a well known result on derivation of an integral group ring to -derivation on it for a finite group G with some conditions on σ and τ. In the process of the extension, a generalization of an application of Skolem-Noether Theorem to derivation on a finite dimensional central simple algebra has also been given for the -derivation case.

Primitivity of group rings of non-elementary torsion-free hyperbolic groups

We use a recent result of Alexander and Nishinaka to show that if G is a non-elementary torsion-free hyperbolic group and R is a countable domain, then the group ring RG is primitive. This implies that the group ring KG of any non-elementary torsion-free hyperbolic group G over a field K is primitive.

Primary group rings

Let R be an associative ring with identity and let J(R) denote the Jacobson radical of R . We say that R is primary if R/J(R) is simple Artinian and J(R) is nilpotent. In this paper we obtain necessary and sufficient conditions for the group ring RG , where G is a nontrivial abelian group, to be primary.

R-Clean Group Rings

Let R be an associative ring with unity. An element a ∈ R is said to be r-clean if a = e + r, where e is an idempotent and r is a von Neumann regular element in R. If every element of R is r-clean, then R is called an r-clean ring. In this paper, we investigate the conditions under which the group ring RG is r-clean. We show that if R is a ring and G is a locally finite p-group with p ∈ J(R), then the group ring RG is r-clean if and only if R is r-clean.

Commutative neat group rings

ABSTRACTA 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.

The Subgroup Normalizer Problem for Integral Group Rings of Some Nilpotent and Metacyclic Groups

For a group G and a subgroup H of G, this article discusses the normalizer of H in the units of a group ring RG. We prove that H is only normalized by the “obvious” units, namely products of elements of G normalizing H and units of RG centralizing H, provided H is cyclic. Moreover, we show that the normalizers of all subgroups of certain nilpotent and metacyclic groups in the corresponding group rings are as small as possible. These classes contain all dihedral groups, all finite nilpotent groups, and all finite groups with all Sylow subgroups being cyclic.