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Articles 1 - 16 of 16
Full-Text Articles in Physical Sciences and Mathematics
Corner’S Theorem On Modules With Anti-Isomorphic Endomorphism Algebras, Brendan Goldsmith, Noel White
Corner’S Theorem On Modules With Anti-Isomorphic Endomorphism Algebras, Brendan Goldsmith, Noel White
Articles
We present a version of an unpublished result of A.L.S. Corner on p-adic modules with anti-isomorphic endomorphism algebras. The result gives a complete description of necessary conditions for two such modules to have anti-isomorphic endomorphism algebras and a sufficient condition is also given. A main difference in the current version is that extensive use is made of our ability to describe certain homomorphism groups.
Algebraic Entropies, Hopficity And Co-Hopficity Of Direct Sums Of Abelian Groups, Brendan Goldsmith, Katao Kong
Algebraic Entropies, Hopficity And Co-Hopficity Of Direct Sums Of Abelian Groups, Brendan Goldsmith, Katao Kong
Articles
Necessary and sufficient conditions to ensure that the direct sum of two Abelian groups with zero entropy is again of zero entropy are still unknown; interestingly the same problem is also unresolved for direct sums of Hopfian and co-Hopfian groups. We obtain sufficient conditions in some situations by placing restrictions on the homomorphisms between the groups. There are clear similarities between the various cases but there is not a simple duality involved.
R-Hopfian And L-Co-Hopfian Abelian Groups, Brendan Goldsmith, Katao Gong
R-Hopfian And L-Co-Hopfian Abelian Groups, Brendan Goldsmith, Katao Gong
Articles
The notions of Hopfian and co-Hopfian groups are well known in both non-commutative and Abelian group theory. In this work we begin a systematic investigation of natural generalizations of these concepts and, in the case of Abelian p-groups, give a complete characterization of the generalizations in terms of the original concepts.
On Projection-Invariant Subgroups Of Abelian P-Groups, Brendan Goldsmith
On Projection-Invariant Subgroups Of Abelian P-Groups, Brendan Goldsmith
Articles
A subgroup P of an Abelian p-group G is said to be projection-invariant in G if Pf is contained in P for all idempotent endomorphisms f. Clearly fully invariant subgroups are projection invariant, but the converse is not true in general. Hausen and Megibben have shown that in many familiar situations these two concepts coincide. In a different direction, the authors have previously introduced the notions of socle-regular and strongly socle-regular groups by focussing on the socles of fully invariant and characteristic subgroups of p-groups. In the present work the authors examine the socles of projection-invariant subgroups of Abelian p-groups.
On Super And Hereditarily Hopfian And Co-Hopfian Abelian Groups, Brendan Goldsmith, Katao Kong
On Super And Hereditarily Hopfian And Co-Hopfian Abelian Groups, Brendan Goldsmith, Katao Kong
Articles
The notions of Hopfian and co-Hopfian groups have been of interest for some time. In this present work we characterize the more restricted classes of hereditarily Hopfian (co-Hopfian) and super Hopfian (co-Hopfian) groups in the case where the groups are Abelian.
On Adjoint Entropy Of Abelian Groups, Brendan Goldsmith, Ketao Gong
On Adjoint Entropy Of Abelian Groups, Brendan Goldsmith, Ketao Gong
Articles
The theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. In the present work we introduce a \lq dual\rq \ notion based upon the replacement of the finite groups used in the definition of algebraic entropy, by subgroups of finite index. The basic properties of this new entropy are established and a connection to Hopfian groups is investigated.
The Maximal Pure Spectrum Of An Abelian Group, Brendan Goldsmith, Ruediger Goebel
The Maximal Pure Spectrum Of An Abelian Group, Brendan Goldsmith, Ruediger Goebel
Articles
This paper introduces the notion of the maximal pure spectrum of an Abelian group - this is the set of isomorphism classes of maximal proper pure subgroups - and focuses on the situation in which this spectrum is small. The converse situation is also examined i.e. given a collection of isomorphism classes of groups, can one find an Abelian group having precisely this collection as its maximal pure spectrum. Finally, it is shown that in some familiar situations, the answers to these questions may be undecidable.
On Modules Which Are Self-Slender, R. Gobel, Brendan Goldsmith, O. Kolman
On Modules Which Are Self-Slender, R. Gobel, Brendan Goldsmith, O. Kolman
Articles
This paper is an examination of the dual of the fundamental isomorphism relating homomorphism groups involving direct sums and direct products over arbitrary index sets. Recall that a module G is said to be self-slender if every homomorphism from a countable product of copies of G into G, vanishes on all but finitely many of the components of the product. Modules of this type are investigated. The simplest version of the results obtained is that under weak cardinality restrictions, there exist non-slender but self-slender Abelian groups.
Algebraic Entropy For Abelian Groups, Dikran Dikranjan, Brendan Goldsmith, Luigi Salce, Paolo Zanardo
Algebraic Entropy For Abelian Groups, Dikran Dikranjan, Brendan Goldsmith, Luigi Salce, Paolo Zanardo
Articles
The theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. Here we study the algebraic entropy of the endomorphisms of Abelian groups, introduced in 1965 by Adler, Konheim and McAndrew. The so-called Addition Theorem is proved; this expresses the algebraic entropy of an endomorphism $ \phi$ of a torsion group as the sum of the algebraic entropies of the restriction to a $ \phi$-invariant subgroup and of the endomorphism induced on the quotient group. Particular attention is paid to endomorphisms with zero algebraic entropy …
A Note On Clean Abelian Groups, Brendan Goldsmith, P. Vamos
A Note On Clean Abelian Groups, Brendan Goldsmith, P. Vamos
Articles
Nicholson defined a ring to be clean if every element is the sum of a unit and an idempotent. A module is clean if its endomorphism algebra is clean. We show that torsion-complete Abelian p-groups are clean and characterize the clean groups among the class of totally projective p-groups. An example is given of a clean p-group which is neither totally projective nor torsion- complete
On Cosmall Abelian Groups, Brendan Goldsmith, O. Kolman
On Cosmall Abelian Groups, Brendan Goldsmith, O. Kolman
Articles
It is a well-known homological fact that every Abelian groupGhas the property that Hom(G,−)com-mutes with direct products. Here we investigate the ‘dual’ property: an Abelian groupGis said to be cosmallif Hom(−,G)commutes with direct products. We show that cosmall groups are cotorsion-free and that nogroup of cardinality less than a strongly compact cardinal can be cosmall. In particular, if there is a properclass of strongly compact cardinals, then there are no cosmall group
Aspects Of Minimality In Abelian Groups, Seosamh Ó Hógáin
Aspects Of Minimality In Abelian Groups, Seosamh Ó Hógáin
Doctoral
This thesis investigates those abelian groups which are minimal with respect to certain quasi-orders defined on Abk, the category of abelian groups of a given infinite cardinality k. Six such quasi-orders are defined and groups which are minimal with respect to these quasi-orders are called either quasi-minimal, with the associated concepts of purely and directly quasi-minimal groups, or simple minimal with the corresponding associated groups. A complete characterisation is derived for the quasi-minimal groups and, assuming GCH, for the purely quasi-minimal groups. Moreover, it is shown that the direct quasi-minimality of a group may be undecidable in ZFC. In the …
Maximal Order Abelian Subgroups Of Symmetric Groups, J. M. Burns, Brendan Goldsmith
Maximal Order Abelian Subgroups Of Symmetric Groups, J. M. Burns, Brendan Goldsmith
Articles
No abstract available
A Note On Coslender Groups, R. Dimitric, Brendan Goldsmith
A Note On Coslender Groups, R. Dimitric, Brendan Goldsmith
Articles
The notion of a coslender group has been introduced previously by the first author. This work continues the investigation of such groups, defines coslender part of a group, proves embeddability results, gives a characterization of finite rank coslender grops, and proves a result on smooth ascending chains of coslender groups with a conjecture that every countable coslender torsion free group is a smooth ascending union of finite rank coslender pure subgroups.
An Essentially Semi-Rigid Class Of Modules, Brendan Goldsmith
An Essentially Semi-Rigid Class Of Modules, Brendan Goldsmith
Articles
No abstract available
A Note On Elongations Of Abelian Groups, Brendan Goldsmith
A Note On Elongations Of Abelian Groups, Brendan Goldsmith
Articles
No abstract available.