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).

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,...

In this paper, we introduce related comparability for exchange ideals. Let $I$ be an exchange ideal of a ring $R$. If $I$ satisfies related comparability, then for any regular matrix $A\in M_n\left(I\right)$, there exist left invertible $U_1,U_2\in M_n\left(R\right)$ and right invertible $V_1,V_2\in M_n\left(R\right)$ such that $U_1{V}_{1}A{U}_2{V}_2=diag(e_1,\cdots ,e_n)$ for idempotents $e_1,\cdots ,e_n\in I$.

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).

Let $R$ be an exchange ring in which all regular elements are one-sided unit-regular. Then every regular element in $R$ is the sum of an idempotent and a one-sided unit. Furthermore, we extend this result to exchange rings satisfying related comparability.

In this paper we investigate the related comparability over exchange rings. It is shown that an exchange ring R satisfies the related comparability if and only if for any regular x C R, there exists a related unit w C R and a group G in R such that wx C G.

We characterize exchange rings having stable range one. An exchange ring $R$ has stable range one if and only if for any regular $a\in R$, there exist an $e\in E\left(R\right)$ and a $u\in U\left(R\right)$ such that $a=e+u$ and $aR\cap eR=0$ if and only if for any regular $a\in R$, there exist $e\in r.ann\left(a^{+}\right)$ and $u\in U\left(R\right)$ such that $a=e+u$ if and only if for any $a,b\in R$, $R/aR\cong R/bR⟹aR\cong bR$.

It is shown that a ring $R$ is a $GM$-ring if and only if there exists a complete orthogonal set $\{e_1,\cdots ,e_n\}$ of idempotents such that all $e_{i}R{e}_i$ are $GM$-rings. We also investigate $GM$-rings for Morita contexts, module extensions and power series rings.

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].

We give a full description of locally finite $2$-groups $G$ such that the normalized group of units of the group algebra $FG$ over a field $F$ of characteristic $2$ has exponent $4$.

Given a field K of characteristic p > 2 and a finite group G, necessary and sufficient conditions for the unit group U(KG) of the group algebra KG to be centrally metabelian are obtained. It is observed that U(KG) is centrally metabelian if and only if KG is Lie centrally metabelian.