Mollov on a decomposition formula in commutative modular group rings (Proceedings of the Plovdiv University-Math., 1973). <...> Introduction Traditionally, suppose RG is the group ring (often regarded as an R-algebra) of an abelian group G over a commutative ring with identity of prime characteristic, for instance, p. <...> As usual, V (RG) denotes the group of normalized units in RG, and S(RG) is its Sylow p-component. <...> For an abelian group G, the letter Gp will denote its p-torsion part and for a commutative unitary ring R, the letter N(R) denotes its nil-radical. <...> In closing, we correct a proof by Mollov [11] in a rather special case. <...> Likewise, we demonstrate that some assertions from [13] are not original and can be simplified in a more convenient form not as they stand. 2007 Danchev P. V. c 2–4 Danchev P. V. 2. <...> Direct Factors and Decompositions in Modular Abelian Group Rings Our point of departure here is to proceed by proving the following two main assertions, motivated the writing up of the paper presented. <...> Suppose G is an abelian group with proper subgroups A and B, and suppose R is a commutative ring with unity of prime characteristic p. <...> Suppose that G is an abelian group with proper subgroups A and B, and suppose that R is a commutative ring with unity of prime characteristic p. <...> The incorrectness arises from the fact that Mollov [12], without any concrete arguments, said that our proof of a statement from [1] is incorrect. <...> In [9] we have shown that the Mollov’s sentence is really inadequate. <...> To avoid such purported reviewer’s reports in the future, we shortly represent some trivial, but crucial, facts concerning the inner construction of the basis group members from the canonical record of any element of the group algebra. <...> What we need to argue is that G = AB holds under the additional restrictions that Ap ⊆ Bp or N(R) = 0, because the other ratio Gp = ApBp plus the converse part half were done in [9]. <...> Since by what we have already shown in [9], gp ∈ ApBp (even gp ∈ AB is completely enough), via the previous commentary it is only <...>
Владикавказский_математический_журнал_№2_2007.pdf
Р О С С И Й С К А Я А К А Д Е М И Я Н А У К
В Л А Д И К А В К А З С К И Й Н А У Ч Н Ы Й Ц Е Н Т Р
ИНСТИТУТ ПРИКЛАДНОЙ МАТЕМАТИКИ И ИНФОРМАТИКИ
ВЛАДИКАВКАЗСКИЙ
МАТЕМАТИЧЕСКИЙ ЖУРНАЛ
Том 9, Выпуск 2
апрель–июнь, 2007
СОДЕРЖАНИЕ
Danchev P. V. On a decomposition equality in modular group rings . . . . . . . . . . . . . . . . 3
Кондаков В. П. О дифференцируемости отображений и строении
пространств голоморфных функций на бесконечномерных пространствах . . . . 9
Kusraev A. G. When all separately band preserving bilinear operators are
symmetric? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Хубежты И. А. Решение проблемы Аргунова — Глисона . . . . . . . . . . . . . . . . . . . . . . . 26
Чилин В. И., Ганиев И. Г., Кудайбергенов К. К. ГНС-представление
C∗-алгебр над кольцом измеримых функций . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Шамоян Р. Ф. Характеризации типа ВМО, диагональное отображение и
ограниченнность интегральных операторов в некоторых пространствах
аналитических функций . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Шубарин М. А. Условия интерполяционности для семейств пространств
Фреше . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Владикавказ
2007
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Владикавказский математический журнал
апрель–июнь, 2007, Том 9, Выпуск 2
UDC 512.742
ON A DECOMPOSITION EQUALITY IN MODULAR GROUP RINGS
P. V. Danchev
Let G be an abelian group such that A G with p-component Ap and B G, and let R be a commutative
ring with 1 of prime characteristic p with nil-radical N(R). It is proved that if Ap ⊆ Bp or N(R) = 0,
then S(RG) = S(RA)(1 + Ip(RG;B)) ⇐⇒ G = AB and Gp = ApBp. In particular, if Ap = 1 or
N(R) = 0, then S(RG) = S(RA) Ч (1 + Ip(RG;B)) ⇐⇒ G = A Ч B. So, the question concerning
the validity of this formula is completely exhausted. The main statement encompasses both the results
of this type established by the author in (Hokkaido Math. J., 2000) and (Miskolc Math. Notes, 2005).
We also point out and eliminate in a concrete situation an error in the proof of a statement due to T.
Zh. Mollov on a decomposition formula in commutative modular group rings (Proceedings of the Plovdiv
University-Math., 1973).
Mathematics Subject Classification (2000): 16S34, 16U60, 20K10, 20K21.
Key words: direct factors, decompositions, normed unit groups, homomorphisms.
1. Introduction
Traditionally, suppose RG is the group ring (often regarded as an R-algebra) of an abelian
group G over a commutative ring with identity of prime characteristic, for instance, p.
As usual, V (RG) denotes the group of normalized units in RG, and S(RG) is its Sylow
p-component. For any subgroup C of G, the symbol I(RG;C) will designate the relative
augmentation ideal of RG with respect to C, and Ip(RG;C) designates its nil-radical.
For an abelian group G, the letter Gp will denote its p-torsion part and for a commutative
unitary ring R, the letter N(R) denotes its nil-radical.
In a series of our investigations (e. g. [1–4, 6, 7]), we study how the direct decomposition
of G can be translated on S(RG); another treatment but of Gp was demonstrated in [5].
In [9] we generalized the foregoing results of this direction by considering an ordinary, not
necessarily direct, decomposition and by exploring how such a decomposing of S(RG) implies
a corresponding one of G and Gp.
The aim of this article is to strengthen the aforementioned results from [9] and to settle
completely the existence of such a decomposition formula for S(RG). In doing that, we use
some helpful facts that are of independent interest as well.
In closing, we correct a proof by Mollov [11] in a rather special case. The generalized
version is wide-open yet. Likewise, we demonstrate that some assertions from [13] are not
original and can be simplified in a more convenient form not as they stand.
2007 Danchev P. V.
c
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