COMMUTATIVE GROUP ALGEBRAS OF ABELIAN GROUPS WITH UNCOUNTABLE POWERS AND LENGTHS

Let F be a field of char(F) = p &#62; 0 and G an abelian group with p-component Gp of cardinality at most &#8501;1 and length at most &#969;1. The main affirmation on the Direct Factor Problem is that S(FG)/Gp is totally projective whenever F is perfect. This extends results due to May (Contemp. Math., 1989) and Hill-Ullery (Proc. Amer. Math. Soc., 1990). As applications to the Isomorphism Problem, suppose that for any group H the F-isomorphism FH &#8773; FG holds. Then if Gp is totally projective, Hp &#8773; Gp. This partially solves a problem posed by May (Proc. Amer. Math. Soc., 1988). In particular, H &#8773; G provided G is p-mixed of torsion-free rank one so that Gp is totally projective. The same isomorphism H &#8773; G is fulfilled when G is p-local algebraically compact too. Besides if Fp is the simple field with p-elements and Gp FpH is a coproduct of torsion complete groups, FpH &#8773; FpG as Fp Fp-algebras implies Hp &#8773; Gp. This expands the central theorem obtained by us in (Rend. Sem. Mat. Univ. Padova, 1999) and partly settles the generalized version of a question raised by May (Proc. Amer. Math. Soc.,1979) as well. As a consequence, when Gp is torsion complete and G is p-mixed of torsion-free rank one, H &#8773; G. Moreover, if G is a coproduct of p-local algebraically compact groups then H &#8773; G. The last attainment enlarges an assertion of Beers-Richman-Walker (Rend. Sem. Mat. Univ. Padova, 1983). Each of the reported achievements strengthens our statements in this direction (Southeast Asian Bull. Math., 2001-2002) and also continues own studies in this aspect (Hokkaido Math. J., 2000) and (Kyungpook Math. J., 2004).</p

Countable extensions of torsion Abelian groups

summary: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

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 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