Let $g\mapsto g^*$ denote an involution on a group $G$. For any (commutative, associative) ring $R$ (with $1$), $*$ extends linearly to an involution of the group ring $RG$. An element $\alpha\in RG$ is symmetric if $\alpha^*=\alpha$ and skew-symmetric if $\alpha^*=-\alpha$. The skew-symmetric elements are closed under the Lie bracket, $[\alpha,\beta]=\alpha\beta-\beta\alpha$. In this paper, we investigate when this set is also closed under the ring product in $RG$. The symmetric elements are closed under the Jordan product, $\alpha\circ\beta=\alpha\beta+\beta\alpha$. Here, we determine when this product is trivial. These two problems are analogues of problems about the skew-symmetric and symmetric elements in group rings that have received a lot of attention.

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.

Let $KG$ be a non-commutative strongly Lie solvable group algebra of a group $G$ over a field $K$ of positive characteristic $p$. In this note we state necessary and sufficient conditions so that the strong Lie derived length of $KG$ assumes its minimal value, namely $\lceil \log_{2}(p+1)\rceil $.

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.

We study group algebras $FG$ which can be graded by a finite abelian group $\Gamma$ such that $FG$ is $\beta$-commutative for a skew-symmetric bicharacter $\beta$ on $\Gamma$ with values in $F^*$.

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

A counterexample is given to a conjecture of Ikeda by finding a class of Gorenstein rings of embedding dimension $3$ with larger Dilworth number than Sperner number. The Dilworth number of $A[Z/pZ\oplus Z/pZ]$ is computed when $A$ is an unramified principal Artin local ring.

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.

Let $R$ be a ring with $1$ and $P(R)$ the periodic radical of $R$. We obtain necessary and sufficient conditions for $P(\RG) = 0$ when $\RG$ is the group ring of an $\FC$ group $G$ and $R$ is commutative. We also obtain a complete description of $P\bigl(I (X, R)\bigr)$ when $I(X,R)$ is the incidence algebra of a locally finite partially ordered set $X$ and $R$ is commutative.

Let $G$ be any group, and $H$ be a normal subgroup of $G$. Then M.~Hartl identified the subgroup $G \cap(1+\triangle^3(G)+\triangle(G)\triangle(H))$ of $G$. In this note we give an independent proof of the result of Hartl, and we identify two subgroups $G\cap(1+\triangle(H)\triangle(G)\triangle(H)+\triangle([H,G])\triangle(H))$, $G\cap(1+\triangle^2(G)\triangle(H)+\triangle(K)\triangle(H))$ of $G$ for some subgroup $K$ of $G$ containing $[H,G]$.