Nilpotent-by-Noetherian Factorized Groups

1989 ◽  
Vol 32 (4) ◽  
pp. 391-403
Author(s):  
Bernhard Amberg ◽  
Silvana Franciosi ◽  
Francesco de Giovanni

AbstractIt is shown that a soluble-by-finite product G = AB of a nilpotent-by-noetherian group A and a noetherian group B is nilpotentby- noetherian. Moreover, a bound for the torsion-free rank of the Fitting factor group of G is given, in terms of the torsion-free rank of the Fitting factor group of A and the torsion-free rank of B.

2010 ◽  
Vol 09 (05) ◽  
pp. 825-837 ◽  
Author(s):  
PAUL BAGINSKI ◽  
ROSS KRAVITZ

Let M be a Krull monoid. Then every element of M may be written as a finite product of irreducible elements. If for every a ∈ M, each two factorizations of a have the same number of irreducible elements, then M is called half-factorial. Using a property of element exponentiation, we provide a new characterization of half-factoriality, valid for all Krull monoids whose class group has torsion-free rank at most one.


2010 ◽  
Vol 03 (02) ◽  
pp. 275-293
Author(s):  
Peter Danchev

Suppose FG is the F-group algebra of an arbitrary multiplicative abelian group G with p-component of torsion Gp over a field F of char (F) = p ≠ 0. Our theorems state thus: The factor-group S(FG)/Gp of all normed p-units in FG modulo Gp is always totally projective, provided G is a coproduct of groups whose p-components are of countable length and F is perfect. Moreover, if G is a p-mixed coproduct of groups with torsion parts of countable length and FH ≅ FG as F-algebras, then there is a totally projective p-group T of length ≤ Ω such that H × T ≅ G × T. These are generalizations to results by Hill-Ullery (1997). As a consequence, if G is a p-splitting coproduct of groups each of which has p-component with length < Ω and FH ≅ FG are F-isomorphic, then H is p-splitting. This is an extension of a result of May (1989). Our applications are the following: Let G be p-mixed algebraically compact or p-mixed splitting with torsion-complete Gp or p-mixed of torsion-free rank one with torsion-complete Gp. Then the F-isomorphism FH ≅ FG for any group H implies H ≅ G. Moreover, letting G be a coproduct of torsion-complete p-groups or G be a coproduct of p-local algebraically compact groups, then [Formula: see text]-isomorphism [Formula: see text] for an arbitrary group H over the simple field [Formula: see text] of p-elements yields H ≅ G. These completely settle in a more general form a question raised by May (1979) for p-torsion groups and also strengthen results due to Beers-Richman-Walker (1983).


2018 ◽  
Vol 61 (1) ◽  
pp. 295-304 ◽  
Author(s):  
R. R. Andruszkiewicz ◽  
M. Woronowicz

AbstractThe relation between the structure of a ring and the structure of its additive group is studied in the context of some recent results in additive groups of mixed rings. Namely, the notion of the square subgroup of an abelian group, which is a generalization of the concept of nil-group, is considered mainly for mixed non-splitting abelian groups which are the additive groups only of rings whose all subrings are ideals. A non-trivial construction of such a group of finite torsion-free rank no less than two, for which the quotient group modulo the square subgroup is not a nil-group, is given. In particular, a new class of abelian group for which an old problem posed by Stratton and Webb has a negative solution, is indicated. A new, far from obvious, application of rings in which the relation of being an ideal is transitive, is obtained.


2017 ◽  
Vol 219 (2) ◽  
pp. 817-834 ◽  
Author(s):  
Martin R. Bridson ◽  
Dessislava H. Kochloukova

Author(s):  
Radosław Łukasik

AbstractLet X be a Banach space. Fix a torsion-free commutative and cancellative semigroup S whose torsion-free rank is the same as the density of $$X^{**}$$ X ∗ ∗ . We then show that X is complemented in $$X^{**}$$ X ∗ ∗ if and only if there exists an invariant mean $$M:\ell _\infty (S,X)\rightarrow X$$ M : ℓ ∞ ( S , X ) → X . This improves upon previous results due to Bustos Domecq (J Math Anal Appl 275(2):512–520, 2002), Kania (J Math Anal Appl 445:797–802, 2017), Goucher and Kania (Studia Math 260:91–101, 2021).


1979 ◽  
Vol 27 (3) ◽  
pp. 284-288 ◽  
Author(s):  
Robert O. Stanton

AbstractLet N be a direct summand of a module which is a direct sum of modules of torsion-free rank one over a discrete valuation ring. Then there is a torsion module T such that N⊕T is also a direct sum of modules of torsion-free rank one.


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