Let R be a ring with identity of characteristic two and G a nontrivial torsion group. We show that if the units in the group ring $RG$ are all trivial, then G must be cyclic of order two or three. We also consider the case where R is a commutative ring with identity of odd prime characteristic and G is a nontrivial locally finite group. We show that in this case, if the units in $RG$ are all trivial, then G must be cyclic of order two. These results improve on a result of Herman et al. [‘Trivial units for group rings with G-adapted coefficient rings’, Canad. Math. Bull.48(1) (2005), 80–89].

An element a in a ring R is left annihilator-stable (or left AS) if, whenever $Ra+{\rm l}(b)=R$ with $b\in R$ , $a-u\in {\rm l}(b)$ for a unit u in R, and the ring R is a left AS ring if each of its elements is left AS. In this paper, we show that the left AS elements in a ring form a multiplicatively closed set, giving an affirmative answer to a question of Nicholson [J. Pure Appl. Alg.221 (2017), 2557–2572.]. This result is used to obtain a necessary and sufficient condition for a formal triangular matrix ring to be left AS. As an application, we provide examples of left AS rings R over which the triangular matrix rings ${\mathbb T}_n(R)$ are not left AS for all $n\ge 2$ . These examples give a negative answer to another question of Nicholson [J. Pure Appl. Alg.221 (2017), 2557–2572.] whether R/J(R) being left AS implies that R is left AS.

Let F be a field of characteristic two and G a finite abelian 2-group with an involutory automorphism η. If G = H × D with non-trivial subgroups H and D of G such that η inverts the elements of H (H without a direct factor of order 2) and fixes D element-wise, then the linear extension of η to the group algebra FG is called a nice involution. This determines the groups of unitary and symmetric normalized units of FG. We calculate the orders and the invariants of these subgroups.

In this paper we study the behavior of the first Zassenhaus conjecture (ZC1) under direct products, as well as the General Bovdi Problem (Gen-BP), which turns out to be a slightly weaker variant of (ZC1). Among other things, we prove that (Gen-BP) holds for Sylow tower groups, and so in particular for the class of supersolvable groups.

(ZC1) is established for a direct product of Sylow-by-abelian groups provided the normal Sylow subgroups form together a Hall subgroup. We also show (ZC1) for certain direct products with one of the factors a Frobenius group.

We extend the classical HeLP method to group rings with coefficients from any ring of algebraic integers. This is used to study (ZC1) for the direct product $G\times A$, where $A$ is a finite abelian group and $G$ has order at most 95. For most of these groups we show that (ZC1) is valid and for all of them that (Gen-BP) holds. Moreover, we also prove that (Gen-BP) holds for the direct product of a Frobenius group with any finite abelian group.

Let $R$ be a ring. A map $f\,:\,R\,\to \,R$ is additive if $f(a\,+\,b)\,=\,f(a)\,+\,f(b)$ for all elements $a$ and $b$ of $R$. Here, a map $f\,:\,R\,\to \,R$ is called unit-additive if $f(u\,+\,v)\,=\,f(u)\,+\,f(v)$ for all units $u$ and $v$ of $R$. Motivated by a recent result of $\text{Xu}$, $\text{Pei}$ and $\text{Yi}$ showing that, for any field $F$, every unit-additive map of ${{\mathbb{M}}_{n}}(F)$ is additive for all $n\,\ge \,2$, this paper is about the question of when every unit-additivemap of a ring is additive. It is proved that every unit-additivemap of a semilocal ring $R$ is additive if and only if either $R$ has no homomorphic image isomorphic to ${{\mathbb{Z}}_{2}}\,\text{or}\,R/J(R)\,\cong \,{{\mathbb{Z}}_{2}}\,$ with $2\,=\,0$ in $R$. Consequently, for any semilocal ring $R$, every unit-additive map of ${{\mathbb{M}}_{n}}(R)$ is additive for all $n\,\ge \,2$. These results are further extended to rings $R$ such that $R/J(R)$ is a direct product of exchange rings with primitive factors Artinian. A unit-additive map $f$ of a ring $R$ is called unithomomorphic if $f(uv)\,=\,f(u)f(v)$ for all units $u$, $v$ of $R$. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.

Let $R$ be a ring. The following results are proved. $\left( 1 \right)$ Every element of $R$ is a sum of an idempotent and a tripotent that commute if and only if $R$ has the identity ${{x}^{6}}\,=\,{{x}^{4}}$ if and only if $R\,\cong \,{{R}_{1}}\,\times \,{{R}_{2}}$, where ${{{R}_{1}}}/{J\left( {{R}_{1}} \right)}\;$ is Boolean with $U\left( {{R}_{1}} \right)$ a group of exponent $2$ and ${{R}_{2}}$ is zero or a subdirect product of ${{\mathbb{Z}}_{3}}^{,}s$. $\left( 2 \right)$ Every element of $R$ is either a sum or a difference of two commuting idempotents if and only if $R\,\cong \,{{R}_{1}}\,\times \,{{R}_{2}}$, where ${{{R}_{1}}}/{J\left( {{R}_{1}} \right)}\;$ is Boolean with $J\left( R \right)\,=\,0$ or $J\left( R \right)\,=\,\left\{ 0,\,2 \right\}$ and ${{R}_{2}}$ is zero or a subdirect product of ${{\mathbb{Z}}_{3}}^{,}s$. $\left( 3 \right)$ Every element of $R$ is a sum of two commuting tripotents if and only if $R\,\cong \,{{R}_{1}}\,\times \,{{R}_{2}}\,\times \,{{R}_{3}}$, where ${{{R}_{1}}}/{J\left( {{R}_{1}} \right)}\;$ is Boolean with $U\left( {{R}_{1}} \right)$ a group of exponent $2$ , ${{R}_{2}}$ is zero or a subdirect product of ${{\mathbb{Z}}_{3}}^{,}s$, and ${{R}_{3}}$ is zero or a subdirect product of ${{\mathbb{Z}}_{5}}^{,}s$.

The unitary Cayley graph of a ring $R$, denoted $\Gamma \left( R \right)$, is the simple graph defined on all elements of $R$, and where two vertices $x$ and $y$ are adjacent if and only if $x\,-\,y$ is a unit in $R$. The largest distance between all pairs of vertices of a graph $G$ is called the diameter of $G$ and is denoted by $\text{diam}\left( G \right)$. It is proved that for each integer $n\,\ge \,1$, there exists a ring $R$ such that $\text{diam}\left( \Gamma \left( R \right) \right)=n$. We also show that $\text{diam}\left( \Gamma \left( R \right) \right)\in \left\{ 1,2,3,\infty \right\}$ for a ring $R$ with ${R}/{J\left( R \right)}\;$ self-injective and classify all those rings with $\text{diam}\left( \Gamma \left( R \right) \right)\,=\,1,\,2,\,3$, and $\infty$, respectively.

Let $RG$ denote the group ring of the group $G$ over the ring $R$. Using an isomorphism between $RG$ and a certain ring of $n\,\times \,n$ matrices in conjunction with other techniques, the structure of the unit group of the group algebra of the dihedral group of order 8 over any finite field of chracteristic 2 is determined in terms of split extensions of cyclic groups.

A p-group is called powerful if every commutator is a product of pth powers when p is odd and a product of fourth powers when p=2. In the group algebra of a group G of p-power order over a finite field of characteristic p, the group of normalized units is always a p-group. We prove that it is never powerful except, of course, when G is abelian.

It is shown that any torsion unit of the integral group ring $\mathbb{Z}G$ of a finite group $G$ is rationally conjugate to an element of $\pm G$ if $G=XA$ with $A$ a cyclic normal subgroup of $G$ and $X$ an abelian group (thus confirming a conjecture of Zassenhaus for this particular class of groups, which comprises the class of metacyclic groups).

Let F be a field of characteristic p and G a group containing at least one element of order p. It is proved that the group of units of the group algebra FG is a bounded Engel group if and only if FG is a bounded Engel algebra, and that this is the case if and only if G is nilpotent and has a normal subgroup H such that both the factor group G/H and the commutator subgroup H′ are finite p–groups.

For each finite group $G$ for which the integral group ring $\mathbb{Z}G$ has only trivial units, we give ring-theoretic conditions for a commutative ring $R$ under which the group ring $RG$ has nontrivial units. Several examples of rings satisfying the conditions and rings not satisfying the conditions are given. In addition, we extend a well-known result for fields by showing that if $R$ is a ring of finite characteristic and $RG$ has only trivial units, then $G$ has order at most 3.

Let p be a prime, a field of pn elements, and G a finite p-group. It is shown here that if G has a quotient whose commutator subgroup is of order p and whose centre has index pk, then the group of normalized units in the group algebra has a conjugacy class of pnk elements. This was first proved by A. Bovdi and C. Polcino Milies for the case k = 2; their argument is now generalized and simplified. It remains an intriguing question whether the cardinality of the smallest noncentral conjugacy class can always be recognized from this test.

We show that an $\text{RA}$ loop has a torsion-free normal complement in the loop of normalized units of its integral loop ring. We also investigate whether an $\text{RA}$ loop can be normal in its unit loop. Over fields, this can never happen.

Let $G$ be an arbitrary group and let $U$ be a subgroup of the normalized units in $\mathbb{Z}G$. We show that if $U$ contains $G$ as a subgroup of finite index, then $U\,=\,G$. This result can be used to give an alternative proof of a recent result of Marciniak and Sehgal on units in the integral group ring of a crystallographic group.

In this paper, we first show that the central height of the unit group of the integral group ring of a periodic group is at most 2. We then give a complete characterization of the $n$-centre of that unit group. The $n$-centre of the unit group is either the centre or the second centre (for $n\,\ge \,2$).

Let $G$ be a free product of cyclic groups of prime order. The structure of the unit group $U(\mathbb{Q}G)$ of the rational group ring $\mathbb{Q}G$ is given in terms of free products and amalgamated free products of groups. As an application, all finite subgroups of $U(\mathbb{Q}G)$, up to conjugacy, are described and the Zassenhaus Conjecture for finite subgroups in $\mathbb{Z}G$ is proved. A strong version of the Tits Alternative for $U(\mathbb{Q}G)$ is obtained as a corollary of the structural result.

A p-subgroup version of the conjecture of Zassenhaus is proved for some finite solvable groups including solvable groups in which any Sylow p-subgroup is either abelian or generalized quaternion, solvable Frobenius groups, nilpotent-by-nilpotent groups and solvable groups whose orders are not divisible by the fourth power of any prime.

We show that p-groups of maximal class and order p5 are determined by their group algebras over the field of p elements. The most important information requisite for the proof is obtained from a detailed study of the unit group of a quotient algebra of the group algebra, larger than the small group algebra.

Special cases of Bovdi's conjecture are proved. In particular the conjecture is proved for supersolvable and Frobenius groups. We also prove that if is finite, α ∊ VℤG a torsion unit and m the smallest positive integer such that αm ∊ G then m divides .