Let $G$ be an abelian group, $R$ a commutative ring of prime characteristic $p$ with identity and $R_{t}G$ a commutative twisted group ring of $G$ over $R$. Suppose $p$ is a fixed prime, $G_p$ and $S\left(R_{t}G\right)$ are the $p$-components of $G$ and of the unit group $U\left(R_{t}G\right)$ of $R_{t}G$, respectively. Let $R^{*}$ be the multiplicative group of $R$ and let $f_{\alpha }\left(S\right)$ be the $\alpha$-th Ulm-Kaplansky invariant of $S\left(R_{t}G\right)$ where $\alpha$ is any ordinal. In the paper the invariants $f_n\left(S\right)$, $n\in \mathbb{N}\cup \left\{0\right\}$, are calculated, provided $G_p=1$. Further, a commutative ring $R$ with identity of prime characteristic $p$ is said...

It is proved that if $A$ is an abelian $p$-group with a pure subgroup $G$ so that $A/G$ is at most countable and $G$ is either $p^{\omega +n}$-totally projective or $p^{\omega +n}$-summable, then $A$ is either $p^{\omega +n}$-totally projective or $p^{\omega +n}$-summable as well. Moreover, if in addition $G$ is nice in $A$, then $G$ being either strongly $p^{\omega +n}$-totally projective or strongly $p^{\omega +n}$-summable implies that so is $A$. This generalizes a classical result of Wallace (J. Algebra, 1971) for totally projective $p$-groups as well as continues our recent investigations in (Arch....

A necessary and sufficient condition is given for the direct sum of two ${ℬ}^{\left(1\right)}$-groups to be (quasi-isomorphic to) a ${ℬ}^{\left(1\right)}$-group. A ${ℬ}^{\left(1\right)}$-group is a torsionfree Abelian group that can be realized as the quotient of a finite direct sum of rank 1 groups modulo a pure subgroup of rank 1.

$B^{\left(1\right)}$-groups are a class of torsionfree Abelian groups of finite rank, part of the main class of Butler groups. In the paper C. Metelli, On direct sums of $B^{\left(1\right)}$-groups, Comment. Math. Univ. Carolinae 34 (1993), 587–591, the problem of direct sums of $B^{\left(1\right)}$-groups was discussed, and a necessary and sufficient condition was given for the direct sum of two $B^{\left(1\right)}$-groups to be a $B^{\left(1\right)}$-group. While sufficiency holds, necessity was wrongly claimed; we solve here the problem, and in the process study a curious hierarchy among...

An elementary proof is given for Hutchinson's duality theorem, which states that if a lattice identity λ holds in all submodule lattices of modules over a ring R with unit element then so does the dual of λ.