scholarly journals ON PROJECTIVELY INERT SUBGROUPS OF COMPLETELY DECOMPOSABLE FINITE RANK GROUPS

2020 ◽  
pp. 63-68
Author(s):  
Andrey R. CHEKHLOV ◽  
◽  
Olesya V. IVANETS ◽  
Keyword(s):  
Prime Number ◽  
Finite Rank ◽  
Invariant Subgroup ◽  
Direct Sum ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

Let a group G be a finite direct sum of torsion-free rank 1 groups Gi. It is proved that every projectively inert subgroup of G is commensurate with a fully invariant subgroup if and only if all Gi are not divisible by any prime number p, and for different subgroups Gi and Gj their types are either equal or incomparable.

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.

scholarly journals Spectra of conjugated ideals in group algebras of abelian groups of finite rank and control theorems

1996 ◽  
Vol 38 (3) ◽  
pp. 309-320 ◽  
Author(s):  
Anatolii V. Tushev
Keyword(s):  
Finite Rank ◽  
Torsion Group ◽  
Group Algebras ◽  
Prime Ideals ◽  
Locally Finite ◽  
Torsion Free Group ◽  
Free Rank ◽  
And Control ◽  
Torsion Free Rank ◽  
Torsion Free

Throughout kwill denote a field. If a group Γ acts on aset A we say an element is Γ-orbital if its orbit is finite and write ΔΓ(A) for the subset of such elements. Let I be anideal of a group algebra kA; we denote by I+ the normal subgrou(I+1)∩A of A. A subgroup B of an abelian torsion-free group A is said to be dense in A if A/B is a torsion-group. Let I be an ideal of a commutative ring K; then the spectrum Sp(I) of I is the set of all prime ideals P of K such that I≤P. If R is a ring, M is an R-module and x ɛ M we denote by the annihilator of x in R. We recall that a group Γ is said to have finite torsion-free rank if it has a finite series in which each factoris either infinite cyclic or locally finite; its torsion-free rank r0(Γ) is then defined to be the number of infinite cyclic factors in such a series.

scholarly journals Finite torsion-free rank endomorphism rings

2015 ◽  
Vol 31 (1) ◽  
pp. 39-43
Author(s):  
SIMION BREAZ ◽  
Keyword(s):  
Abelian Group ◽  
Small Group ◽  
Endomorphism Ring ◽  
Endomorphism Rings ◽  
Direct Sum ◽  
Bounded Group ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

We prove that a finite torsion-free rank abelian group with finite torsion-free rank endomorphism ring is a direct sum of a bounded group and a self-small group.

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).

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