EUDML

We show the inheritance of summable property for certain fully invariant subgroups by the whole group and vice versa. The results are somewhat parallel to these due to Linton (Mich. Math. J., 1975) and Linton-Megibben (Proc. Amer. Math. Soc., 1977). They also generalize recent assertions of ours in (Alg. Colloq., 2009) and (Bull. Allah. Math. Soc., 2008)

Isomorphism of modular group algebras of direct sums of torsion-complete abelian $p$ -groups

Peter V. Danchev — 1999

Rendiconti del Seminario Matematico della Università di Padova

Warfield invariants in abelian group rings.

Peter V. Danchev — 2005

Extracta Mathematicae

Let R be a perfect commutative unital ring without zero divisors of (R) = p and let G be a multiplicative abelian group. Then the Warfield p-invariants of the normed unit group V (RG) are computed only in terms of R and G. These cardinal-to-ordinal functions, combined with the Ulm-Kaplansky p-invariants, completely determine the structure of V (RG) whenever G is a Warfield p-mixed group.

Invo-regular unital rings

Peter V. Danchev — 2018

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

It was asked by Nicholson (Comm. Algebra, 1999) whether or not unit-regular rings are themselves strongly clean. Although they are clean as proved by Camillo-Khurana (Comm. Algebra, 2001), recently Nielsen and Ster showed in Trans. Amer. Math. Soc., 2018 that there exists a unit-regular ring which is not strongly clean. However, we define here a proper subclass of rings of the class of unit-regular rings, called invo-regular rings, and establish that they are strongly clean. Interestingly, without...

Basic subgroups in abelian group rings

Peter Vassilev Danchev — 2002

Czechoslovak Mathematical Journal

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 $U_{p} (R G)$ and of the factor-group $U_{p} (R G) / G$ of the unit group $U (R G)$ in the modular group algebra $R G$ are established, in the case when $R$ is weakly perfect. Moreover, a lower basic subgroup and a basic subgroup of the normed $p$ -component $S (R G)$ and of the quotient group $S (R G) / G_{p}$ are given when $R$ is perfect and $G$ is arbitrary whose $G / G_{p}$ is $p$ -divisible....

The generalized criterion of Dieudonné for primary valuated groups

Peter Vassilev Danchev — 2007

Acta Mathematica Universitatis Ostraviensis

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).

The generalized criterion of Dieudonné for valuated $p$ -groups

Peter Vassilev Danchev — 2006

Acta Mathematica Universitatis Ostraviensis

We prove that if $G$ is an abelian $p$ -group with a nice subgroup $A$ so that $G / A$ is a $Σ$ -group, then $G$ is a $Σ$ -group if and only if $A$ is a $Σ$ -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 $Σ$ -group precisely when $A$ is a $Σ$ -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....

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,...

Commutative group algebras of highly torsion-complete abelian $p$ -groups

Peter Vassilev Danchev — 2003

Commentationes Mathematicae Universitatis Carolinae

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).

$G$ -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].

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 $G \in 𝕂$ , a class of abelian groups, does imply that $A \in 𝕂$ . 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.

A note on the countable extensions of separable $p^{ω + n}$ -projective abelian $p$ -groups

Peter Vassilev Danchev — 2006

Archivum Mathematicum

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

Page 1 Next

Download Results (CSV)

Advanced Search

Formula preview

Currently displaying 1 – 20 of 28

A note on the isomorphism of modular group algebras of $p$ -mixed Abelian groups with divisible $p$ -components.

Nice bases for mixed and torsion-free Abelian groups.

The isomorphic description of torsion units in commutative group rings over fields with positive characteristic.

A note on isomorphic commutative group algebras over certain rings.

On isomorphic commutative group algebras of Abelian groups with $p^{ω + n}$ -projective quotients.

Warfield Invariants in Abelian Group Algebras

Fully Invariant Subgroups of $n$ -Summable Primary Abelian Groups

On Some Fully Invariant Subgroups of Summable Groups

Isomorphism of modular group algebras of direct sums of torsion-complete abelian $p$ -groups

Warfield invariants in abelian group rings.

Invo-regular unital rings

Basic subgroups in abelian group rings

The generalized criterion of Dieudonné for primary valuated groups

Direct decompositions and basic subgroups in commutative group rings

The generalized criterion of Dieudonné for valuated $p$ -groups

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

Commutative group algebras of highly torsion-complete abelian $p$ -groups

$G$ -nilpotent units of commutative group rings

Countable extensions of torsion Abelian groups

A note on the countable extensions of separable $p^{ω + n}$ -projective abelian $p$ -groups