THE TORSION FREE RANK

2014 ◽  
pp. 187-199
Keyword(s):  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

On the Square Subgroup of a Mixed SI-Group

2018 ◽  
Vol 61 (1) ◽  
pp. 295-304 ◽  
Author(s):  
R. R. Andruszkiewicz ◽  
M. Woronowicz
Keyword(s):  
Abelian Group ◽  
Quotient Group ◽  
Abelian Groups ◽  
Additive Group ◽  
Negative Solution ◽  
New Class ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free ◽  
Group Modulo

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.

scholarly journals The torsion-free rank of homology in towers of soluble pro-p groups

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

scholarly journals Invariant vector means and complementability of Banach spaces in their second duals

2022 ◽  
Author(s):  
Radosław Łukasik
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Studia Math ◽  
Cancellative Semigroup ◽  
Invariant Mean ◽  
Invariant Vector ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

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).

scholarly journals Decompositions of modules over a discrete valuation ring

1979 ◽  
Vol 27 (3) ◽  
pp. 284-288 ◽  
Author(s):  
Robert O. Stanton
Keyword(s):  
Direct Summand ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Torsion Module ◽  
Direct Sum ◽  
Rank One ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

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.

QUASI-ISOMORPHISMS AND GROUPS OF QUASI-HOMOMORPHISMS

2009 ◽  
Vol 08 (05) ◽  
pp. 617-627
Author(s):  
ULRICH ALBRECHT ◽  
SIMION BREAZ
Keyword(s):  
Abelian Group ◽  
Dual Problem ◽  
Abelian Groups ◽  
Mixed Abelian Group ◽  
Free Rank ◽  
Mixed Groups ◽  
Torsion Free Rank ◽  
Torsion Free

This paper investigates to which extent a self-small mixed Abelian group G of finite torsion-free rank is determined by the groups Hom (G,C) where C is chosen from a suitable class [Formula: see text] of Abelian groups. We show that G is determined up to quasi-isomorphism if [Formula: see text] is the class of all self-small mixed groups C with r0(C) ≤ r0(G). Several related results are given, and the dual problem of orthogonal classes is investigated.

scholarly journals Localization in non-noetherian group rings

1980 ◽  
Vol 21 (1) ◽  
pp. 151-163 ◽  
Author(s):  
P. F. Smith
Keyword(s):  
Abelian Group ◽  
Prime Ideal ◽  
Noetherian Ring ◽  
General Setting ◽  
Group Rings ◽  
Regular Local Ring ◽  
Characteristic P ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

Let k be a field and G an Abelian group of finite torsion-free rank. Brewer, Costa and Lady [1, Theorem A] showed that if k has characteristic 0 then each Localization of the group algebra kG at a prime ideal is a regular local ring. They also showed (in the same theorem) that if k has characteristic p>0, then kG is locally Joetherian (i.e. each localization of kG at a prime ideal is a Noetherian ring) if and only if G is an extension of a finitely generated group by a torsion p′-group. The purpose of this note is to examine this theorem in a more general setting.

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