scholarly journals Full subrings of E-rings

1996 ◽  
Vol 54 (2) ◽  
pp. 275-280
Author(s):  
Shalom Feigelstock
Keyword(s):  
Additive Group ◽  
Left Multiplication ◽  
Ring Of Endomorphisms ◽  
Torsion Free

A ring R is said to be an E-ring if the map R → of E (R)+ into the ring of endomorphisms of its additive group via a ↪ al = left multiplication by a, is an isomorphism. In this note torsion free rings R for which the group Rl, of left multiplication maps by elements of R, is a full subgroup of E(R)+ will be considered. These rings are called TE-rings. It will be shown that TE-rings satisfy many properties of E-rings, and that unital TE-rings are E-rings. If R is a TE-ring, then E(R+) is an E-ring, and E(R+)+ / is bounded. Some results concerning additive groups of TE-rings will be obtained.

scholarly journals Eigenvalues and strong orbit equivalence

2015 ◽  
Vol 36 (8) ◽  
pp. 2419-2440 ◽  
Author(s):  
MARÍA ISABEL CORTEZ ◽  
FABIEN DURAND ◽  
SAMUEL PETITE
Keyword(s):  
Equivalence Class ◽  
Quotient Group ◽  
Additive Group ◽  
Orbit Equivalence ◽  
Dimension Group ◽  
Torsion Free

We give conditions on the subgroups of the circle to be realized as the subgroups of eigenvalues of minimal Cantor systems belonging to a determined strong orbit equivalence class. Actually, the additive group of continuous eigenvalues $E(X,T)$ of the minimal Cantor system $(X,T)$ is a subgroup of the intersection $I(X,T)$ of all the images of the dimension group by its traces. We show, whenever the infinitesimal subgroup of the dimension group associated with $(X,T)$ is trivial, the quotient group $I(X,T)/E(X,T)$ is torsion free. We give examples with non-trivial infinitesimal subgroups where this property fails. We also provide some realization results.

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 Endomorphism rings of Butler groups

1987 ◽  
Vol 42 (3) ◽  
pp. 322-329 ◽  
Author(s):  
D. M. Arnold ◽  
C. I. Vinsonhaler
Keyword(s):  
Additive Group ◽  
Division Algebras ◽  
Endomorphism Rings ◽  
Dimensional Algebra ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Butler Group ◽  
Group A ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractThis note is devoted to the question of deciding whether or not a subring of a finite-dimensional algebra over the rationals, with additive group a Butler group, is the endomorphism ring of a Butler group (a Butler group is a pure subgroup of a finite direct sum of rank-1 torsion-free abelian groups). A complete answer is given for subrings of division algebras. Several applications are included.

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∞).

scholarly journals Groups whose automorphisms are almost determined by their restriction to a subgroup

1986 ◽  
Vol 28 (1) ◽  
pp. 87-93 ◽  
Author(s):  
Martin R. Pettet
Keyword(s):  
Abelian Groups ◽  
Additive Group ◽  
Free Abelian Groups ◽  
Torsion Free

The trivial observation that every automorphism of a group is determined by its restriction to a set of generators suggests the converse question: if X is a subset of a group G such that each automorphism of G is determined (or “almost” determined) by its restriction to X, to what extent is the structure of G governed by that of the subgroup which X generates? Is this subgroup in some sense necessarily “large” in G? If the index of the subgroup is used as a measure of largeness, then in the absence of additional hypotheses, the answer to the second question is generally “no”, the additive group of rationals with X = {1} being an obvious counterexample. (More confounding is the existence of uncountable torsion-free abelian groups for which inversion is the only non-trivial automorphism. See, for example, [2], [3], and [4].) However, under certain finiteness assumptions, it seems that some positive conclusions are obtainable. One such example will be considered here.

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 Sumset and Inverse Sumset Theory for Shannon Entropy

2010 ◽  
Vol 19 (4) ◽  
pp. 603-639 ◽  
Author(s):  
TERENCE TAO
Keyword(s):  
Shannon Entropy ◽  
Difference Sets ◽  
Additive Group ◽  
Random Variable ◽  
Discrete Random Variable ◽  
Finite Sets ◽  
Finite Set ◽  
Torsion Free ◽  
Sharp Lower Bound ◽  
Bounded Rank

Let G = (G, +) be an additive group. The sumset theory of Plünnecke and Ruzsa gives several relations between the size of sumsets A + B of finite sets A, B, and related objects such as iterated sumsets kA and difference sets A − B, while the inverse sumset theory of Freiman, Ruzsa, and others characterizes those finite sets A for which A + A is small. In this paper we establish analogous results in which the finite set A ⊂ G is replaced by a discrete random variable X taking values in G, and the cardinality |A| is replaced by the Shannon entropy H(X). In particular, we classify those random variables X which have small doubling in the sense that H(X1 + X2) = H(X) + O(1) when X1, X2 are independent copies of X, by showing that they factorize as X = U + Z, where U is uniformly distributed on a coset progression of bounded rank, and H(Z) = O(1).When G is torsion-free, we also establish the sharp lower bound $\Ent(X+X) \geq \Ent(X) + \frac{1}{2} \log 2 - o(1)$, where o(1) goes to zero as H(X) → ∞.

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.

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