scholarly journals A Note on Additive Groups of Some Specific Associative Rings

2016 ◽  
Vol 30 (1) ◽  
pp. 219-229
Author(s):  
Mateusz Woronowicz
Keyword(s):  
Associative Ring ◽  
Abelian Groups ◽  
Free Groups ◽  
Additive Group ◽  
Direct Sums ◽  
Rank One ◽  
Elementary Methods ◽  
Associative Rings ◽  
Torsion Free

AbstractAlmost complete description of abelian groups (A, +, 0) such that every associative ring R with the additive group A satisfies the condition: every subgroup of A is an ideal of R, is given. Some new results for SR-groups in the case of associative rings are also achieved. The characterization of abelian torsion-free groups of rank one and their direct sums which are not nil-groups is complemented using only elementary methods.

scholarly journals Automorphisms of nilpotent groups of class two with small rank

1985 ◽  
Vol 39 (1) ◽  
pp. 121-131 ◽  
Author(s):  
Thomas A. Fournelle
Keyword(s):  
Automorphism Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Factor Group ◽  
Direct Sums ◽  
Matrix Computations ◽  
Central Factor ◽  
Rank One ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractRational abelian groups, that is, torsion-free abelian groups of rank one, are characterized by their types. This paper characterizes rational nilpotent groups of class two, that is, nilpotent groups of class two in which the center and central factor group are direct sums of rational abelian groups. This characterization is done according to the types of the summands of the center and the central factor group. Using these types and some cohomological techniques it is possible to determine the automorphism group of the nilpotent group in question by performing essentially matrix computations.In particular, the automorphism groups of rational nilpotent groups of class two and rank three are completely described. Specific examples are given of semicomplete and pseudocomplete nilpotent groups.

SELF-SMALL ABELIAN GROUPS

2009 ◽  
Vol 80 (2) ◽  
pp. 205-216 ◽  
Author(s):  
ULRICH ALBRECHT ◽  
SIMION BREAZ ◽  
WILLIAM WICKLESS
Keyword(s):  
Small Groups ◽  
Abelian Groups ◽  
Direct Sums ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

AbstractThis paper investigates self-small abelian groups of finite torsion-free rank. We obtain a new characterization of infinite self-small groups. In addition, self-small groups of torsion-free rank 1 and their finite direct sums are discussed.

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.

E-Associative Rings

1993 ◽  
Vol 36 (2) ◽  
pp. 147-153 ◽  
Author(s):  
Shalom Feigelstock
Keyword(s):  
Associative Ring ◽  
Additive Group ◽  
Associative Rings ◽  
Torsion Free

AbstractA ring R is E-associative if φ(xy) = φ(x)y for all endomorphisms φ of the additive group of R, and all x,y ∊ R. Unital E-associative rings are E-rings. The structure of the torsion ideal of an E-associative ring is described completely. The E-associative rings with completely decomposable torsion free additive groups are also classified. Conditions under which E-associative rings are E-rings, and other miscellaneous results are obtained.

scholarly journals The additive group of the rationals does not have an automatic presentation

2011 ◽  
Vol 76 (4) ◽  
pp. 1341-1351 ◽  
Author(s):  
Todor Tsankov
Keyword(s):  
Abelian Groups ◽  
Free Groups ◽  
Additive Group ◽  
Or Groups ◽  
Infinite Set ◽  
Torsion Free

AbstractWe prove that the additive group of the rationals does not have an automatic presentation. The proof also applies to certain other abelian groups, for example, torsion-free groups that are p-divisible for infinitely many primes p, or groups of the form ⊕pϵIZ(p∞), where I is an infinite set of primes.

scholarly journals Locally irreducible rings

1985 ◽  
Vol 32 (1) ◽  
pp. 129-145 ◽  
Author(s):  
C. Vinsonhaler ◽  
W. Wickless
Keyword(s):  
Finite Rank ◽  
Abelian Groups ◽  
Free Groups ◽  
Additive Group ◽  
Infinite Rank ◽  
Field Of Definition ◽  
Free Abelian Groups ◽  
Definition Of ◽  
Torsion Free

In the study of torsion-free abelian groups of finite rank the notions of irreducibility, field of definition and E-ring have played significant rôles. These notions are tied together in the following theorem of R. S. Pierce:THEOREM. Let R be a ring whose additive group is torsion free finite rank irreducible and let Γ be the centralizer of QR as a QE(R) module. Then Γ is the unique smallest field of definition of R. Moreover, Γ ∩ R is an E-ring, in fact, it is a maximal E-subring of R.In this paper we consider extensions of Pierce's result to the infinite rank case. This leads to the concept of local irreducibility for torsion free groups.

Decision problems concerning S-arithmetic groups

1985 ◽  
Vol 50 (3) ◽  
pp. 743-772 ◽  
Author(s):  
Fritz Grunewald ◽  
Daniel Segal
Keyword(s):  
Additive Group ◽  
Nilpotent Groups ◽  
Isomorphism Problem ◽  
Algorithmic Problem ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Finite Dimensional ◽  
Associative Rings ◽  
Finitely Presented ◽  
Torsion Free

This paper is a continuation of our previous work in [12]. The results, and some applications, have been described in the announcement [13]; it may be useful to discuss here, a little more fully, the nature and purpose of this work.We are concerned basically with three kinds of algorithmic problem: (1) isomorphism problems, (2) “orbit problems”, and (3) “effective generation”.(1) Isomorphism problems. Here we have a class of algebraic objects of some kind, and ask: is there a uniform algorithm for deciding whether two arbitrary members of are isomorphic? In most cases, the answer is no: no such algorithm exists. Indeed this has been one of the most notable applications of methods of mathematical logic in algebra (see [26, Chapter IV, §4] for the case where is the class of all finitely presented groups). It turns out, however, that when consists of objects which are in a certain sense “finite-dimensional”, then the isomorphism problem is indeed algorithmically soluble. We gave such algorithms in [12] for the following cases: = {finitely generated nilpotent groups}; = {(not necessarily associative) rings whose additive group is finitely generated}; = {finitely Z-generated modules over a fixed finitely generated ring}.Combining the methods of [12] with his own earlier work, Sarkisian has obtained analogous results with the integers replaced by the rationals: in [20] and [21] he solves the isomorphism problem for radicable torsion-free nilpotent groups of finite rank and for finite-dimensional Q-algebras.

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 Characterization of the automorphisms having the lifting property in the category of abelianp-groups

2003 ◽  
Vol 2003 (71) ◽  
pp. 4511-4516
Author(s):  
S. Abdelalim ◽  
H. Essannouni
Keyword(s):  
Homomorphic Image ◽  
Abelian Groups ◽  
Torsion Group ◽  
Divisible Group ◽  
Torsion Free

Letpbe a prime. It is shown that an automorphismαof an abelianp-groupAlifts to any abelianp-group of whichAis a homomorphic image if and only ifα=π idA, withπan invertiblep-adic integer. It is also shown that ifAis torsion group or torsion-freep-divisible group, thenidAand−idAare the only automorphisms ofAwhich possess the lifting property in the category of abelian groups.

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