Let $S\left(RG\right)$ be a normed Sylow $p$-subgroup in a group ring $RG$ of an abelian group $G$ with $p$-component $G_p$ and a $p$-basic subgroup $B$ over a commutative unitary ring $R$ with prime characteristic $p$. The first central result is that $1+I(RG;B_p)+I(R\left(p^i\right)G;G)$ is basic in $S\left(RG\right)$ and $B[1+I(RG;B_p)+I(R\left(p^i\right)G;G)]$ is $p$-basic in $V\left(RG\right)$, and $[1+I(RG;B_p)+I(R\left(p^i\right)G;G)]G_p/G_p$ is basic in $S\left(RG\right)/G_p$ and $[1+I(RG;B_p)+I(R\left(p^i\right)G;G)]G/G$ is $p$-basic in $V\left(RG\right)/G$, provided in both cases $G/G_p$ is $p$-divisible and $R$ is such that its maximal perfect subring $R^{p^i}$ has no nilpotents whenever $i$ is natural. The second major result is that $B(1+I(RG;B_p))$ is $p$-basic in $V\left(RG\right)$ and $(1+I(RG;B_p))G/G$ is $p$-basic in $V\left(RG\right)/G$,...

Suppose $F$ is a perfect field of $\mathrm{c}harF=p\ne 0$ and $G$ is an arbitrary abelian multiplicative group with a $p$-basic subgroup $B$ and $p$-component $G_p$. Let $FG$ be the group algebra with normed group of all units $V\left(FG\right)$ and its Sylow $p$-subgroup $S\left(FG\right)$, and let $I_p(FG;B)$ be the nilradical of the relative augmentation ideal $I(FG;B)$ of $FG$ with respect to $B$. The main results that motivate this article are that $1+I_p(FG;B)$ is basic in $S\left(FG\right)$, and $B(1+I_p(FG;B))$ is $p$-basic in $V\left(FG\right)$ provided $G$ is $p$-mixed. These achievements extend in some way a result of N. Nachev (1996) in Houston...

A torsion-free group is a $B_2$-group if and only if it has an axiom-3 family $ℭ$ of decent subgroups such that each member of $ℭ$ has such a family, too. Such a family is called $S{L}_{{\aleph }_0}$-family. Further, a version of Shelah’s Singular Compactness having a rather simple proof is presented. As a consequence, a short proof of a result [R1] stating that a torsion-free group $B$ in a prebalanced and TEP exact sequence $0\to K\to C\to B\to 0$ is a $B_2$-group provided $K$ and $C$ are so.