Isomorphism of Commutative Group Algebras over all Fields

It is argued that the commutative group algebra over each field determines up to an isomorphism its group basis for any of the following group classes: • Direct sums of cocyclic groups • Splitting countable modulo torsion groups whose torsion parts are direct sums of cyclics; • Splitting groups whose torsion parts are separable countable • Groups whose torsion parts are algebraically compact • Algebraically compact groups These give a partial positive answer to the R.Brauer’s classical problem

Rings with Jacobson units

We introduce and study the notion of JU rings, that are, rings having only Jacobson units. In parallel to the so-called UU rings, these rings also form a large class and have many interesting properties established in the present paper. For instance, it is proved that any exchange JU ring is semi-boolean, and vice versa. This somewhat extends a result due to Lee-Zhou (Glasg. Math. J., 2008) and Danchev-Lam (Publ. Math. Debrecen, 2016)

On the trivial units in finite commutative group rings

Let G be a finite abelian group and F a finite field. A criterion is found for all units in the group ringFG to be trivial. This attainment is also extended to the general case for arbitrary abelian groups and fields

On exchange π-UU unital rings

We prove that a ring R is exchange 2-UU if, and only if, J(R) is nil and R/J(R)≅B×C, where B is a Boolean ring and C is a ring with C ⊆ Πμ ℤ₃ for some ordinal μ. We thus somewhat improve on a result due to Abdolyousefi-Chen (J. Algebra Appl., 2018) by showing that it is a simple consequence of already well-known results of Danchev-Lam (Publ. Math. Debrecen, 2016) and Danchev (Commun. Korean Math. Soc., 2017)

Basic subgroups in abelian group rings

summary:Suppose $R$ is a commutative ring with identity of prime characteristic $p$ and $G$ is an arbitrary abelian $p$-group. In the present paper, a basic subgroup and a lower basic subgroup of the $p$-component $U_p(RG)$ and of the factor-group $U_p(RG)/G$ of the unit group $U(RG)$ in the modular group algebra $RG$ are established, in the case when $R$ is weakly perfect. Moreover, a lower basic subgroup and a basic subgroup of the normed $p$-component $S(RG)$ and of the quotient group $S(RG)/G_p$ are given when $R$ is perfect and $G$ is arbitrary whose $G/G_p$ is $p$-divisible. These results extend and generalize a result due to Nachev (1996) published in Houston J. Math., when the ring $R$ is perfect and $G$ is $p$-primary. Some other applications in this direction are also obtained for the direct factor problem and for a kind of an arbitrary basic subgroup

Warfield Invariants of V(RG)/GV(RG)/G

Let $R$ be a commutativeunitary ring of prime characteristic $p$ and let $G$ be an Abelian group. We calculateonly in terms of $R$ and $G$ (and their sections)Warfield $p$-invariants of thequotient group $V(RG)/G$, that is, the group of all normalized units $V(RG)$ in the groupring $RG$ modulo $G$. This supplies recent results of ours in (Extr. Math., 2005),(Collect. Math., 2008) and (J. Algebra Appl., 2008)