Narrow Results

A ring $R$ is called right small injective if every homomorphism from a small right ideal of $R$ to $R_R$ can be extended to a homomorphism from $R_R$ to $R_R$. Recall that a right ideal $I$ of $R$ is called small if for any proper right ideal $K$ of $R$, $I+K\ne R$. It is proved that if $G$ is a locally finite group and the group ring $RG$ is right small injective, then $R$ must be right small injective. The converse is not true even if $G$ is a finite group. An example is given to show a right small injective group ring $RG$ whose coefficient ring $R$ is not right small injective. In the end, let $G$ be a finite group, an equivalent characterization is given for $RG$ to be right small injective.

Let F be a non-cyclic free group and R, S its normal subgroups. We study the abelian group , viewed as a module over F/RS, via conjugation in F, and residual nilpotence of the group F/[R, S]. An application to the asphericity of finite presentations is given.

For fixed m, n ≥ 2, we examine the structure of the nth lower central subgroup γ_n(F) of the free group F of rank m with respect to a certain finite chain F = F⁽⁰⁾ > F⁽¹⁾ > ⋯ > F^(l-1) > F^(l) = {1} of free groups in which F^(k) is of finite rank m(k) and is contained in the kth derived subgroup δ_k(F) of F. The derived subgroups δ_k(F/γ_n(F)) of the free nilpotent group F/γ_n(F) are isomorphic to the quotients F^(k)/(F^(k) ∩ γ_n(F)) and admit presentations of the form 〈x_k,1,…,x_k,m(k): γ_(n)(F^(k))〉, where γ_(n)(F^(k)), contained in γ_n(F), is a certain partial lower central subgroup of F^(k). We give a complete description of γ_n(F) as a staggered product Π_{1 ≤ k ≤ l-1}(γ_〈n〉(F^(k))*γ_[n](F^(k)))^F(k+1), where γ_〈n〉(F^(k)) is a free factor of the derived subgroup [F^(k),F^(k)] of F^(k) having countable infinite rank and generated by a certain set of reduced commutators of weight at least n, and γ_[n](F^(k)) is the subgroup generated by a certain finite set of products of non-reduced ordered commutators of weight at least n. There are some far-reaching consequences.

Marciniak and Sehgal showed that if u is a non-trivial bicyclic unit of an integral group ring then there is a bicyclic unit v such that u and v generate a non-abelian free group. A similar result does not hold for Bass cyclic units of infinite order based on non-central elements as some of them have finite order modulo the center. We prove a theorem that suggests that this is the only limitation to obtain a non-abelian free group from a given Bass cyclic unit. More precisely, we prove that if u is a Bass cyclic unit of an integral group ring ℤG of a solvable and finite group G, such that u has infinite order modulo the center of U(ℤG) and it is based on an element of prime order, then there is a non-abelian free group generated by a power of u and a power of a unit in ℤG which is either a Bass cyclic unit or a bicyclic unit.

We study free subgroups in the multiplicative group of division rings generated by group rings of some classes of soluble groups.

Let $G$ be an almost simple group with socle $A_n$, the alternating group of degree $n$. We prove that there is a unit of order $pq$ in the integral group ring of $G$ if and only if there is an element of that order in $G$ provided $p$ and $q$ are primes greater than $\frac{n}{3}$. We combine this with some explicit computations to verify the prime graph question for all almost simple groups with socle $A_n$ if $n\le 17$.

In this work, we extend an established isomorphism between group rings and a subring of the $n\times n$ matrices. This extension allows us to construct more complex matrices over the ring $R.$ We present many interesting examples of complex matrices constructed directly from our extension. We also show that some of the matrices used in the literature before can be obtained by a direct application of our extended isomorphism.

Let G be a torsion group and let R be a G-adapted ring. In this note we study the question of when the group ring RG has only trivial torsion units. It turns out that the above question is closely related to the question of when the quaternion group ring RQ₈ has only trivial torsion units. We first give a ring-theoretic condition on R which determines exactly when the quaternion group ring has only trivial torsion units. Then several equivalent conditions for RG to have only trivial torsion units are provided. We also investigate the hypercenter of the unit group of a G-adapted group ring RG, and show that when R satisfies the torsion trivial involution condition, this hypercenter is not equal to the center if and only if G is a Q*-group.

Let G be a group with cyclic derived subgroup of p-power order (p > 2), and F a field of characteristic p. We determine the derived length of the group of units of the group algebra FG.

In this paper we characterize *-prime group rings. We prove that the group ring RG of the group G over the ring R is *-prime if and only if R is *-prime and Λ⁺(G) = (1). In the process we obtain more examples of group rings which are *-prime but not strongly prime.

In 1996, Jespers and Wang classified finite semigroups whose integral semigroup ring has finitely many units. In a recent paper, Iwaki–Juriaans–Souza Filho continued this line of research by partially classifying the finite semigroups whose rational semigroup algebra contains a ℤ-order with hyperbolic unit group. In this paper, we complete this classification and give an easy proof that deals with all finite semigroups.

Let J(R) be the Jacobson radical of a ring R. Then R is called homogeneous semilocal if R/J(R) is simple artinian. The aim of this paper is to find necessary and sufficient conditions for the group rings and the crossed products to be homogeneous semilocal ring.

In this survey we revise the methods and results on the existence and construction of free groups of units in group rings, with special emphasis in integral group rings over finite groups and group algebras. We also survey results on constructions of free groups generated by elements which are either symmetric or unitary with respect to some involution and other results on which integral group rings have large subgroups which can be constructed with free subgroups and natural group operations.

We prove there is no ring with unit group isomorphic to S_n for n ≥ 5 and that there is no ring with unit group isomorphic to A_n for n ≥ 5, n ≠ 8. To prove the non-existence of such a ring, we prove the non-existence of a certain ideal in the group algebra 𝔽₂[G], with G an alternating or symmetric group as above. We also give examples of rings with unit groups isomorphic to S₁, S₂, S₃, S₄, A₁, A₂, A₃, A₄, and A₈. Most of our existence results are well-known, and we recall them only briefly; however, we expect the construction of a ring with unit group isomorphic to S₄ to be new, and so we treat it in detail.

When considering the unit group of ${{𝒪}}_{F}G$ (${{𝒪}}_F$ the ring of integers of an abelian number field $F$ and a finite group $G$) certain components in the Wedderburn decomposition of $FG$ cause problems for known generic constructions of units; these components are called exceptional. Exceptional components are divided into two types: type 1 is division rings, type 2 is $2\times 2$-matrix rings. For exceptional components of type 1 we provide infinite classes of division rings by describing the seven cases of minimal groups (with respect to quotients) having those division rings in their Wedderburn decomposition over $F$. We also classify the exceptional components of type 2 appearing in group algebras of a finite group over number fields $F$ by describing all 58 finite groups $G$ having a faithful exceptional Wedderburn component of this type in $FG$.

A Schur ring $\mathcal{𝒜}$ is called Dedekind if the formal sum of every $\mathcal{𝒜}$-subgroup is in the center of $\mathcal{𝒜}$. In this paper, we find all finite groups $G$ such that every proper Schur ring over $G$ is Dedekind. As a corollary of our main theorem, we find all finite groups $G$ such that every proper Schur ring over $G$ is commutative.

A ring $R$ is called semiboolean if $R/J(R)$ is boolean and idempotents lift modulo $J(R)$, where $J(R)$ denotes the Jacobson radical of $R$. In this paper, we define $J$-boolean rings as a generalization of semiboolean rings. A ring $R$ is said to be J-boolean if $R/J(R)$ is boolean. Various basic properties of these rings are obtained. The $J$-boolean group rings and skew group rings have been studied. It is investigated whether the results obtained for $J$-boolean group rings also hold for the skew group rings.

Recall that a ring $R$ is called right ${\aleph }_0$-injective if every homomorphism from a countably generated right ideal of $R$ to $R_R$ can be extended to a homomorphism from $R_R$ to $R_R$. These rings are not only a natural generalization of self-injective rings but also strongly connected with regularities of rings. Let $RG$ be the group ring of a group $G$ over a ring $R$. It is proved that $RG$ is right ${\aleph }_0$-injective if and only if (i) $R$ is right ${\aleph }_0$-injective; (ii) $G$ is finite; (iii) for each countably generated right ideal $I$ of $RG$, any right $R$-homomorphism from $I_R$ to $R_R$ can be extended to a right $R$-homomorphism from $R{G}_R$ to $R_R$.

In this paper, we obtain the structure of the unit groups $U(F{Q}_{2^n})$ of the finite group algebras of the generalized quaternion groups $Q_{2^n}$, over a finite field $F$ of characteristic $p>2$.