Group Ring RG Research Articles

Cohomology of groups and splendid equivalences of derived categories

In an earlier paper we studied the impact of equivalences between derived categories of group rings on their cohomology rings. Especially the group of auto-equivalences TrPic(RG) of the derived category of a group ring RG as introduced by Raphaël Rouquier and the author defines an action on the cohomology ring of this group. We study this action with respect to the restriction map, transfer, conjugation and the local structure of the group G.

Local analysis of the normalizer problem

For a finite group G, and a commutative ring R, the automorphisms of G inducing an inner automorphism of the group ring RG form a group Aut R ( G). Let Aut int( G)=Aut A ( G), where A is the ring of all algebraic integers in C . It is shown how Clifford theory can be used to analyze Aut int( G). It is proved that Aut int( G)/Inn( G) is an abelian group, and can indeed be any finite abelian group. It is an outstanding question whether Aut Z (G)= Inn(G) if G has an abelian Sylow 2-subgroup. This is shown to be true in some special cases, but also a group G with abelian Sylow subgroups and Aut int( G)≠Inn( G) is given.

On exponents and Auslander-Reiten components of irreducible lattices

Let G be a finite group, and let R be a complete discrete rank one valuation ring of characteristic zero with maximal ideal \(\max (R) = \pi R\), and residue class field \(R/\pi R\) of characteristic p > 0. The notion of the exponent of an RG-lattice L is due to J. F. Carlson and the first author [1]. In this note we use it to show that any non-projective absolutely irreducible RG-lattice L with indecomposable factor module \(\bar {L} = L/\pi L\) lies at the end of its connected component \(\Theta \) of the stable Auslander-Reiten quiver \(\Gamma _s(RG)\) of the group ring RG. Since such lattices L belong to p-blocks B with non-trivial defect groups \(\delta (B)\) we also study some relations between the order of \(\delta (B)\) and the exponent exp(L).

The Group Ring of SL2(pf) over p-adic Integers for p Odd

Let p>2 be a prime, R=Zp[ζpf−1], K=Qp[ζpf−1], and G=SL2(pf). The group ring RG is calculated nearly up to Morita equivalence: The projections of RG into the simple components of KG are given explicitly and the endomorphism rings and homomorphism bimodules between the projective indecomposable RG-lattices are described.

Projective Representations and Relative Semisimplicity

Let R be a local (commutative) ring and let p be a prime not invertible in R. Let G be a finite group of order divisible by p. It is well known that the group ring RG admits nonprojective lattices (e.g., R itself with the trivial action). For any element α∈H2(G,R*) one can form the twisted group ring RαG. The “twisting problem” asks whether there exists a class α s.t. the corresponding twisted group ring admits only projective lattices. For fields of characteristic p, the answer is in E. Aljadeff and D. J. S. Robinson [J. Pure Appl. Algebra94 (1994), 1–15]. Here we answer this question for rings of the form Zps, s≥2. The main tools are the classification of modular representation of the Klein 4 group over Z2 and a Chouinard-like theorem [E. Aljadeff and Y. Ginosar, J. Algebra179 (1996), 599–606] for twisted group rings.

The Normalizer of a Group in the Unit Group of Its Group Ring

Let G be a group and R a commutative ring. One of the main objects Ž . investigated in the theory of group rings is the unit group U s U R, G of the group ring RG. Despite many efforts, the structure of this group is very poorly understood, especially when the group is infinite. The aim of Ž . this work is to study the normalizer N s N R, G of G in U. This group acts by conjugation on G, and the group of automorphisms of G obtained in this way is denoted by Aut G. Therefore Aut G is isomorphic to NrC, R R Ž . where C s C R, G is the center of U. Since G : N, the group Inn G of inner automorphisms of G is contained in Aut G, and by Out G we R R denote the group Aut GrInn G f NrGC. Out interest in the group N is R motivated by the following properties:

On the regular radical of a group ring

If R is a commutative ring with identity and G a group, it is shown that the regular radical of the group ring RG is trivial if the regular radical of R is trivial. This answers in the affirmative the commutative case of a question in [4]. Further, the regular radical of RG is computed when G is an Abelian group or when G is locally finite and the order of each element of G is a unit in R.

Exceptional group ring automorphisms for some metabelian groups

Let H be a generalized dihedral, semi-dihedral, quaternion, or modular group, and let A = (u, v, w) be a product of three odd order cyclic groups, with (|v|,|w|) = 1. For R a semi-local Dedekind domain of characteristic 0 in which no prime divisor of |H|.|A| is invertible, we prove that there is a semi-direct product G = H × A such that the group ring RG has an exceptional automorphism, i.e. provides a counter-example to a well-known conjecture of Zassenhaus on automorphisms of group rings

Semigroup identities on units of integral group rings

AbstractLet U(RG) be the group of units of a group ring RG over a commutative ring R with 1. We say that a group is an SIT-group if it is an extension of a group which satisfies a semigroup identity by a torsion group. It is a consequence of the main result that if G is torsion and R = Z, then U(RG) is an SIT-group if and only if G is either abelian or a Hamiltonian 2-group. If R is a local ring of characteristic 0 only the first alternative can occur.

Principally Injective Rings

A ring R is called right principally injective if every R-homomorphism from a principal right ideal to R is left multiplication by an element of R. In this paper various properties of these rings are developed, many extending known results. If, in addition, R is semiperfect and has an essential right socle, it is shown: (1) that the right socle equals the left socle, that this is essential on both sides and is finitely generated on the left; (2) that the two singular ideals coincide; and (3) that R admits a Nakayama permutation of its basic idempotents. These rings are a natural generalization of the pseudo-Frobenius rings, and our work extends results of Björk and Rutter. We also answer a question of Camillo about commutative principally injective rings in which every ideal contains a uniform ideal. Finally, we show that if the group ring RG is principally injective then R is principally injective and G is locally finite; and that if R is right selfinjective and G is locally finite then RG is principally injective, extending results of Farkas.

Regular skew group rings

AbstractLet G be a group acting on a ring R. We study the problem of determining necessary and sufficient conditions in order that the skew group ring RG be von Neumann regular. Complete characterizations are given in some particular situations, including the case where all idempotents of R are central. For a regular ring R admitting a G-invariant pseudo-rank function N, with G finite, we obtain a necessary condition for RG being regular in terms of the induced action of G on the N-completion of R.

Invariants of tile sylowp-subgroup of the unit group of a commutative group ring of characteristicp∗

Let R be a commutative ring with identity of prime characteristic p and let G be an abelian group. Denote by S(RG) the Sylow p-subgroup of the group of normalized units (i.e. the units of augmentation 1) of the group ring RG. The following result is established in the paper.1) Let S be a subring of R containing the identity of R. Then the additive factor-group R(p)/S(p) is isomorphic to the multiplicative group R[p]/S[p], whereR(p) = {x ∊ R∣x p = 0} and R[p] = {y ∊ R∣y p = 1}.2) It holdsR(p)/R p (p) ≌ R[p]/R p [p] where R p = {x p ∣ x ∊ R}.Computing of the Ulm-Kaplansky invariants of the group S(RG) is the main result of the paper. A description of the maximal divisible subgroup of S(RG) is given as well

Torsion units in group rings

Let U(RG) be the unit group of the group ring RG. In this paper we study group rings RG whose support elements of every torsion unit are torsion, where R is either the ring of integers Z or a field K.

Localization in a Morita context with applications to fixed rings

In the study of a noetherian ring R, there are two related questions that have played a major role in recent years; one is finding prime ideals of R which satisfy the second layer condition, the other is finding the (second layer) link closed sets of prime ideals of R. In particular these properties are necessary to determine the classical sets of prime ideals of R, which allows one to obtain a well behaved Ore localization (see, for example, [24, 8, 91). One aspect of this problem is the transfer of these properties from the prime spectrum of one ring R to the prime spectrum of another, related ring, A. Jategaonkar [S] has shown that, if R is a commutative noetherian ring, then the group ring RG satisfies the second layer condition whenever G is polycyclic-by-finite. This result was then generalized by Bell [2] to strongly group graded rings. More recently Letzter [lo] has studied the relationship between the prime spectrums of the noetherian rings R and S, when S is module finite over R, showing how the second layer condition and links between prime ideals transfer from one ring to the other. Let T be a right noetherian ring and let e be an idempotent element of T, then it is well known that eTe is a right noetherian ring. In Sections 1 and 2 we examine the transfer of the second layer condition between the prime ideals of T and eTe, as well as the transfer of classical sets of prime ideals between these rings. This enables us to study how these properties of Spec(eTe) translate to Spec(( 1 e) T( 1 e)). In particular, under certain hypotheses, we prove that whenever eTe satisfies the second layer condition, then so does (1 e) T( 1 e) (Corollary 2.6). In addition we are able to show that under the same hypotheses, if every prime ideal of eTe belongs to a finite classical set, then so does every prime ideal of (1 -e) T(l -e). Another way to describe the relationship between T and eTe is via a Morita context (see [ 11). Let R and S be rings, let s W, and R V, be bimodules, and let 6: VOs W-+ R, $: WOR V+ S be bimodule 373 OOZI-8693/91 $3.00

Group rings with $FC$-nilpotent unit groups

Let U(RG) be the unit group of the group ring RG. Groups G such that U(RG) is FC-nilpotent are determined, where R is the ring of integers Z or a field K of characteristic zero.

Virtually-residually- p automorphism groups of group rings

Let R be a ring and p a prime integer. R is said to be p-shaped if R has an identity, is finitely generated, and has an ideal I of p-power index such that I is residually-nilpotent- p. It is shown that if R is p-shaped, and G is a finitely generated, virtually-residually- p group, then the group of automorphisms Aut RG of the group ring RG is virtually-residually- p.

The mod- p cohomology rings of metacyclic groups

Let e : 1→N→G→K→1 be an extension of a finite cyclic group N by a finite cyclic group K, and let R be a commutative ring with 1. Using homological perturbation theory, we construct a free resolution of R over the group ring RG of G in such a way that the resolution reflects the structure of G as an extension of N by K, and we use this resolution to compute the mod p cohomology ring of G for a prime p via the spectral sequence of the extension e.

Isomorphic Group Rings Over Domains

AbstractLet R and S be rings, G and H abelian groups, and RG and SH the goup rings of G and H over R and S respectively. In this note we consider what relations must hold between G and H or between R and S if the group rings RG and SH are isomorphic. For example, it is shown that if R and S are integral domains of characteristic zero, G and H torsion abelian groups such that if G has an element of order p then p is not invertible in R, and RG and SH are isomorphic, then the rings R and 5 are isomorphic and the groups G and H are isomorphic.

On the automorphisms of the group ring of a finitely generated free abelian group

Let R be an associative ring with 1 and G a finitely generated torsion-free abelian group. In this note, we classify all R-automorphisms of the group ring RG. The special case where G is infinite cyclic was previously settled in [8], and our interest in this problem was rekindled by the recent paper of Mehrvarz and Wallace [7], who carried out the classification in the case where R contains a nilpotent prime ideal.

Class groups and automorphism groups of group rings

This paper is a sequel to [2]. A polycyclic-by-finite group G was there called dihedral free if G contains no subgroup isomorphic to 〈b, a:ba = b-1 a2 = 1〉 whose normalizer has finite index in G. It was shown in [2, Theorem F] that, if R is a commutative Noetherian domain, the group ring RG is a prime Noetherian maximal order if and only if R is integrally closed, G is dihedral free, and G has no non-trivial finite normal subgroups. Throughout, R and G will be assumed to satisfy these hypotheses. The main aim of the paper is to study the class group of the maximal order RG.