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Commutative modular group algebras of p -mixed and p -splitting abelian Σ -groups

Peter Vassilev Danchev (2002)

Commentationes Mathematicae Universitatis Carolinae

Let G be a p -mixed abelian group and R is a commutative perfect integral domain of char R = p > 0 . Then, the first main result is that the group of all normalized invertible elements V ( R G ) is a Σ -group if and only if G is a Σ -group. In particular, the second central result is that if G is a Σ -group, the R -algebras isomorphism R A ≅ R G between the group algebras R A and R G for an arbitrary but fixed group A implies A is a p -mixed abelian Σ -group and even more that the high subgroups of A and G are isomorphic, namely, ℋ A ≅ ℋ G . Besides,...

Countable extensions of torsion Abelian groups

Peter Vassilev Danchev (2005)

Archivum Mathematicum

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 ∈ 𝕂 , a class of abelian groups, does imply that A ∈ 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 𝕂 coincides with the class of all totally projective p -groups.

Direct decompositions and basic subgroups in commutative group rings

Peter Vassilev Danchev (2006)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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).

G -nilpotent units of commutative group rings

Peter Vassilev Danchev (2012)

Commentationes Mathematicae Universitatis Carolinae

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 R G are G -nilpotent. This extends recent results published in [Extracta Math., 2008–2009] and [Ann. Sci. Math. Québec, 2009].

Currently displaying 21 – 40 of 86