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 ${U}_{p}\left(RG\right)$ and of the factor-group ${U}_{p}\left(RG\right)/G$ of the unit group $U\left(RG\right)$ 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 $S\left(RG\right)$ and of the quotient group $S\left(RG\right)/{G}_{p}$ are given when $R$ is perfect and $G$ is arbitrary whose $G/{G}_{p}$ is $p$-divisible....

An attractive interplay between the direct decompositions and the explicit form of basic subgroups in group rings of abelian groups over a commutative unitary ring are established. In particular, as a consequence, we give a simpler confirmation of a more general version of our recent result in this aspect published in Czechoslovak Math. J. (2006).

We prove that if $G$ is an abelian $p$-group with a nice subgroup $A$ so that $G/A$ is a $\Sigma $-group, then $G$ is a $\Sigma $-group if and only if $A$ is a $\Sigma $-subgroup in $G$ provided that $A$ is equipped with a valuation induced by the restricted height function on $G$. In particular, if in addition $A$ is pure in $G$, $G$ is a $\Sigma $-group precisely when $A$ is a $\Sigma $-group. This extends the classical Dieudonné criterion (Portugal. Math., 1952) as well as it supplies our recent results in (Arch. Math. Brno, 2005), (Bull. Math. Soc. Sc. Math....

Let $G$ be a $p$-mixed abelian group and $R$ is a commutative perfect integral domain of $charR=p>0$. Then, the first main result is that the group of all normalized invertible elements $V\left(RG\right)$ is a $\Sigma $-group if and only if $G$ is a $\Sigma $-group. In particular, the second central result is that if $G$ is a $\Sigma $-group, the $R$-algebras isomorphism $RA\cong RG$ between the group algebras $RA$ and $RG$ for an arbitrary but fixed group $A$ implies $A$ is a $p$-mixed abelian $\Sigma $-group and even more that the high subgroups of $A$ and $G$ are isomorphic, namely, ${\mathscr{H}}_{A}\cong {\mathscr{H}}_{G}$. Besides,...

A new class of abelian $p$-groups with all high subgroups isomorphic is defined. Commutative modular and semisimple group algebras over such groups are examined. The results obtained continue our recent statements published in Comment. Math. Univ. Carolinae (2002).

Suppose $R$ is a commutative unital ring and $G$ is an abelian group. We give a general criterion only in terms of $R$ and $G$ when all normalized units in the commutative group ring $RG$ are $G$-nilpotent. This extends recent results published in [Extracta Math., 2008–2009] and [Ann. Sci. Math. Québec, 2009].

Suppose $A$ is an abelian torsion group with a subgroup $G$ such that $A/G$ is countable that is, in other words, $A$ is a torsion countable abelian extension of $G$. A problem of some group-theoretic interest is that of whether $G\in \mathbb{K}$, a class of abelian groups, does imply that $A\in \mathbb{K}$. The aim of the present paper is to settle the question for certain kinds of groups, thus extending a classical result due to Wallace (J. Algebra, 1981) proved when $\mathbb{K}$ coincides with the class of all totally projective $p$-groups.

It is proved that if $G$ is a pure ${p}^{\omega +n}$-projective subgroup of the separable abelian $p$-group $A$ for $n\in N\cup \left\{0\right\}$ such that $|A/G|\le {\aleph}_{0}$, then $A$ is ${p}^{\omega +n}$-projective as well. This generalizes results due to Irwin-Snabb-Cutler (CommentṀathU̇nivṠtṖauli, 1986) and the author (Arch. Math. (Brno), 2005).

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....

We prove that if $G$ is an Abelian $p$-group of length not exceeding $\omega $ and $H$ is its ${p}^{\omega +n}$-projective subgroup for $n\in \mathbb{N}\cup \left\{0\right\}$ such that $G/H$ is countable, then $G$ is also ${p}^{\omega +n}$-projective. This enlarges results of ours in (Arch. Math. (Brno), 2005, 2006 and 2007) as well as a classical result due to Wallace (J. Algebra, 1971).

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...

Suppose $G$ is a subgroup of the reduced abelian $p$-group $A$. The following two dual results are proved: $(*)$ If $A/G$ is countable and $G$ is an almost totally projective group, then $A$ is an almost totally projective group. $(**)$ If $G$ is countable and nice in $A$ such that $A/G$ is an almost totally projective group, then $A$ is an almost totally projective group. These results somewhat strengthen theorems due to Wallace (J. Algebra, 1971) and Hill (Comment. Math. Univ. Carol., 1995), respectively.

Suppose $G$ is a $p$-mixed splitting abelian group and $R$ is a commutative unitary ring of zero characteristic such that the prime number $p$ satisfies $p\notin \text{inv}\left(R\right)\cup \text{zd}\left(R\right)$. Then $R\left(H\right)$ and $R\left(G\right)$ are canonically isomorphic $R$-group algebras for any group $H$ precisely when $H$ and $G$ are isomorphic groups. This statement strengthens results due to W. May published in J. Algebra (1976) and to W. Ullery published in Commun. Algebra (1986), Rocky Mt. J. Math. (1992) and Comment. Math. Univ. Carol. (1995).

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 $V\left(RG\right)/G$,...

We prove that pure subgroups of thick Abelian $p$-groups which are modulo countable are again thick. This generalizes a result due to Megibben (Michigan Math. J. 1966). Some related results are also established.

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