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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....

Basic subgroups in commutative modular group rings

Peter Vassilev Danchev (2004)

Mathematica Bohemica

Let S ( R G ) be a normed Sylow p -subgroup in a group ring R G of an abelian group G with p -component G p and a p -basic subgroup B over a commutative unitary ring R with prime characteristic p . The first central result is that 1 + I ( R G ; B p ) + I ( R ( p i ) G ; G ) is basic in S ( R G ) and B [ 1 + I ( R G ; B p ) + I ( R ( p i ) G ; G ) ] is p -basic in V ( R G ) , and [ 1 + I ( R G ; B p ) + I ( R ( p i ) G ; G ) ] G p / G p is basic in S ( R G ) / G p and [ 1 + I ( R G ; B p ) + I ( R ( p i ) G ; G ) ] G / G is p -basic in V ( R G ) / G , provided in both cases G / G p is p -divisible and R is such that its maximal perfect subring R p i has no nilpotents whenever i is natural. The second major result is that B ( 1 + I ( R G ; B p ) ) is p -basic in V ( R G ) and ( 1 + I ( R G ; B p ) ) G / G is p -basic in V ( R G ) / G ,...

Basic subgroups in modular abelian group algebras

Peter Vassilev Danchev (2007)

Czechoslovak Mathematical Journal

Suppose F is a perfect field of c h a r F = p 0 and G is an arbitrary abelian multiplicative group with a p -basic subgroup B and p -component G p . Let F G be the group algebra with normed group of all units V ( F G ) and its Sylow p -subgroup S ( F G ) , and let I p ( F G ; B ) be the nilradical of the relative augmentation ideal I ( F G ; B ) of F G with respect to B . The main results that motivate this article are that 1 + I p ( F G ; B ) is basic in S ( F G ) , and B ( 1 + I p ( F G ; B ) ) is p -basic in V ( F G ) provided G is p -mixed. These achievements extend in some way a result of N. Nachev (1996) in Houston...

Bernstein sets with algebraic properties

Marcin Kysiak (2009)

Open Mathematics

We construct Bernstein sets in ℝ having some additional algebraic properties. In particular, solving a problem of Kraszewski, Rałowski, Szczepaniak and Żeberski, we construct a Bernstein set which is a < c-covering and improve some other results of Rałowski, Szczepaniak and Żeberski on nonmeasurable sets.

Bethe Ansatz and the geography of rigged strings

Tadeusz Lulek (2007)

Banach Center Publications

We demonstrate the way in which composition of two famous combinatorial bijections, of Robinson-Schensted and Kerov-Kirillov-Reshetikhin, applied to the Heisenberg model of magnetic ring with spin 1/2, defines the geography of rigged strings (which label exact eigenfunctions of the Bethe Ansatz) on the classical configuration space (the set of all positions of the system of r reversed spins). We point out that each l-string originates, in the language of this bijection, from an island of l consecutive...

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