Equivalence elementaire et decidabilite pour des structures du type groupe agissant sur un groupe abelien

1998 ◽  
Vol 63 (4) ◽  
pp. 1255-1285 ◽  
Author(s):  
Patrick Simonetta
Keyword(s):  
Abelian Group ◽  
Soluble Group ◽  
Additive Group ◽  
Transfer Principle ◽  
Divisible Module ◽  
Similar Principle ◽  
Partial Multiplication ◽  
The One ◽  
Invertible Elements ◽  
Torsion Free

AbstractWe prove an Ax-Kochen-Ershov like transfer principle for groups acting on groups. The simplest case is the following: let B be a soluble group acting on an abelian group G so that G is a torsion-free divisible module over the group ring ℤ[B], then the theory of B determines the one of the two-sorted structure 〈G,B,*〉, where * is the action of B on C. More generally, we show a similar principle for structures 〈G,B,*〉, where G is a torsion-free divisible module over the quotient of ℤ[B] by the annulator of G.Two applications come immediately from this result:First, for not necessarily commutative domains, where we consider the action of a subgroup of the invertible elements on the additive group. We obtain then the decidability of a weakened structure of ring, with partial multiplication.The second application is to pure groups. The semi-direct product of G by B is bi-interpretable with our structure 〈G,B,*〉. Thus, we obtain stable decidable groups that are not linear over a field.

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.

BREAKING UP FINITE AUTOMATA PRESENTABLE TORSION-FREE ABELIAN GROUPS

2011 ◽  
Vol 21 (08) ◽  
pp. 1463-1472 ◽  
Author(s):  
GÁBOR BRAUN ◽  
LUTZ STRÜNGMANN
Keyword(s):  
Abelian Group ◽  
Finite Automaton ◽  
Free Abelian Group ◽  
Finite Automata ◽  
Additive Group ◽  
Rational Numbers ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Torsion Free

In [Todor Tsankov, The additive group of the rationals does not have an automatic presentation, May 2009, arXiv:0905.1505v1], it was shown that the group of rational numbers is not FA-presentable, i.e. it does not admit a presentation by a finite automaton. More generally, any torsion-free abelian group that is divisible by infinitely many primes is not of this kind. In this article we extend the result from [13] and prove that any torsion-free FA-presentable abelian group G is an extension of a finite rank free group by a finite direct sum of Prüfer groups ℤ(p∞).

A TORSION-FREE ABELIAN GROUP EXISTS WHOSE QUOTIENT GROUP MODULO THE SQUARE SUBGROUP IS NOT A NIL-GROUP

2016 ◽  
Vol 94 (3) ◽  
pp. 449-456 ◽  
Author(s):  
R. R. ANDRUSZKIEWICZ ◽  
M. WORONOWICZ
Keyword(s):  
Abelian Group ◽  
Quotient Group ◽  
Free Abelian Group ◽  
Free Groups ◽  
Additive Group ◽  
New Method ◽  
Torsion Free Abelian Group ◽  
Elementary Methods ◽  
Torsion Free ◽  
Group Modulo

The first example of a torsion-free abelian group $(A,+,0)$ such that the quotient group of $A$ modulo the square subgroup is not a nil-group is indicated (for both associative and general rings). In particular, the answer to the question posed by Stratton and Webb [‘Abelian groups, nil modulo a subgroup, need not have nil quotient group’, Publ. Math. Debrecen27 (1980), 127–130] is given for torsion-free groups. A new method of constructing indecomposable nil-groups of any rank from $2$ to $2^{\aleph _{0}}$ is presented. Ring multiplications on $p$-pure subgroups of the additive group of the ring of $p$-adic integers are investigated using only elementary methods.

scholarly journals The additive groups of subdirectly irreducible rings

1979 ◽  
Vol 20 (2) ◽  
pp. 165-170 ◽  
Author(s):  
Shalom Feigelstock
Keyword(s):  
Abelian Group ◽  
Additive Group ◽  
Subdirectly Irreducible ◽  
Torsion Free

An abelian group G is said to be subdirectly irreducible if there exists a subdirectly irreducible ring R with additive group G. If G is subdirectly irreducible, and if every ring R with additive group G, and R2 ≠ 0, is subdirectly irreducible, then G is said to be strongly subdirectly irreducible. The torsion, and torsion free, subdirectly irreducible and strongly subdirectly irreducible groups are classified completely. Results are also obtained concerning mixed subdirectly irreducible and strongly subdirectly irreducible groups.

scholarly journals On the additive groups of subdirectly irreducible rings

1986 ◽  
Vol 34 (2) ◽  
pp. 275-281 ◽  
Author(s):  
Yasuyuki Hirano ◽  
Isao Mogami
Keyword(s):  
Abelian Group ◽  
Group Structure ◽  
Additive Group ◽  
Subdirectly Irreducible ◽  
Torsion Free

In this paper we study the additive group structure of subdirectly irreducible rings and their hearts. We give and example of a torsion-free, non-reduced abelian group which is not the underlying additive group of any associative subdirectly irreducible ring. It is a counterexample to a theorem in Feigelstock's book “Additive Groups of Rings.”

scholarly journals Additive groups of rings whose subrings are ideals

1997 ◽  
Vol 55 (3) ◽  
pp. 477-481 ◽  
Author(s):  
Shalom Feigelstock
Keyword(s):  
Abelian Group ◽  
Additive Group ◽  
Torsion Free

An Abelian group G is called an SI-group if for every ring R with additive group R+ = G, every subring S of R is an ideal in R. A complete description is given of the torsion SI-groups, and the completely decomposable torsion free SI-groups. Results are obtained in other cases as well.

On the square subgroups of decomposable torsion-free abelian groups of rank three

2016 ◽  
Vol 7 (4) ◽  
Author(s):  
Fysal Hasani ◽  
Fatemeh Karimi ◽  
Alireza Najafizadeh ◽  
Yousef Sadeghi
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractThe square subgroup of an abelian group

scholarly journals Self-splitting Abelian groups

2001 ◽  
Vol 64 (1) ◽  
pp. 71-79 ◽  
Author(s):  
P. Schultz
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Free Module ◽  
Research Report ◽  
Original Form ◽  
Original Research ◽  
Direct Sum ◽  
The University ◽  
Torsion Free ◽  
Made In

G is reduced torsion-free A belian group such that for every direct sum ⊕G of copies of G, Ext(⊕G, ⊕G) = 0 if and only if G is a free module over a rank 1 ring. For every direct product ΠG of copies of G, Ext(ΠG,ΠG) = 0 if and only if G is cotorsion.This paper began as a Research Report of the Department of Mathematics of the University of Western Australia in 1988, and circulated among members of the Abelian group community. However, it was never submitted for publication. The results have been cited, widely, and since copies of the original research report are no longer available, the paper is presented here in its original form in Sections 1 to 5. In Section 6, I survey the progress that has been made in the topic since 1988.

CANCELLATION LAW AND UNIQUE FACTORIZATION THEOREM FOR STRING OPERATIONS

2006 ◽  
Vol 05 (02) ◽  
pp. 231-243
Author(s):  
DONGVU TONIEN
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Additive Group ◽  
New Product ◽  
Binary String ◽  
Factorization Theorem ◽  
General Criterion ◽  
Unique Factorization ◽  
Product Operation ◽  
Associative Law

Recently, Hoit introduced arithmetic on blocks, which extends the binary string operation by Jacobs and Keane. A string of elements from the Abelian additive group of residues modulo m, (Zm, ⊕), is called an m-block. The set of m-blocks together with Hoit's new product operation form an interesting algebraic structure where associative law and cancellation law hold. A weaker form of unique factorization and criteria for two indecomposable blocks to commute are also proved. In this paper, we extend Hoit's results by replacing the Abelian group (Zm, ⊕) by an arbitrary monoid (A, ◦). The set of strings built up from the alphabet A is denoted by String(A). We extend the operation ◦ on the alphabet set A to the string set String(A). We show that (String(A), ◦) is a monoid if and only if (A, ◦) is a monoid. When (A, ◦) is a group, we prove that stronger versions of a cancellation law and unique factorization hold for (String(A), ◦). A general criterion for two irreducible strings to commute is also presented.

scholarly journals Subgroups of finite index in an additive group of a ring

2001 ◽  
Vol 27 (2) ◽  
pp. 83-89
Author(s):  
Doostali Mojdeh ◽  
S. Hassan Hashemi
Keyword(s):  
Finite Index ◽  
Additive Group ◽  
Infinite Field ◽  
Division Rings ◽  
Invertible Elements

IfKis an infinite field andG⫅Kis a subgroup of finite index in an additive group, thenK∗=G∗G∗−1whereG∗denotes the set of all invertible elements inGandG∗−1denotes all inverses of elements ofG∗. Similar results hold for various fields, division rings and rings.

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