Brendan Goldsmith
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View article: Quantum Arithmetic Algorithms: Implementation, Resource Estimation, and Comparison
Quantum Arithmetic Algorithms: Implementation, Resource Estimation, and Comparison Open
As quantum computing technology advances, the need for optimized arithmetic circuits continues to grow. This paper presents the implementation and resource estimation of a library of quantum arithmetic algorithms, including addition, multi…
View article: On some hereditary and super classes of directly finite Abelian groups
On some hereditary and super classes of directly finite Abelian groups Open
Continuing recent studies of both the hereditary and super properties of certain classes of Abelian groups, we explore in-depth what is the situation in the quite large class consisting of directly finite Abelian groups. Trying to connect …
View article: On Some Hereditary and Super Classes of Directly Finite Abelian Groups
On Some Hereditary and Super Classes of Directly Finite Abelian Groups Open
Continuing recent studies of both the hereditary and super properties of certain classes of Abelian groups, we explore in-depth what is the situation in the quite large class consisting of directly finite Abelian groups. Trying to connect …
View article: Two Generalizations of Hopfian Abelian Groups
Two Generalizations of Hopfian Abelian Groups Open
This paper targets to generalize the notion of Hopfian groups in the commutative case by defining the so-called {\bf relatively Hopfian groups} and {\bf weakly Hopfian groups}, and establishing some their crucial properties and characteriz…
View article: Cyclic subgroup transitivity for Abelian groups
Cyclic subgroup transitivity for Abelian groups Open
In previous work, the first two authors studied the notion of transitivity with respect to cyclic subgroups for separable Abelian p -groups and modules over the ring of p -adic integers. Here we consider briefly how the notion can be used …
View article: Supper Bassian and Nearly Generalized Bassian Abelian Groups
Supper Bassian and Nearly Generalized Bassian Abelian Groups Open
In connection to two recent publications of ours in Arch. Math. Basel (2021) and Acta Math. Hung. (2022), respectively, and in regard to the results obtained in Arch. Math. Basel (2012), we have the motivation to study the near property of…
View article: The Pierce decomposition and Pierce embedding of endomorphism rings of abelian <i>p</i>-groups
The Pierce decomposition and Pierce embedding of endomorphism rings of abelian <i>p</i>-groups Open
The first goal of this paper is to investigate the Pierce decomposition of the endomorphism ring End ( G ) = F ^ ⊕ End s ( G ) {\operatorname{End}(G)=\widehat{F}\oplus\operatorname{End}_{s}(G)} of an abelian p -group …
View article: On endomorphisms and automorphisms of some pure subgroups of the Baer-Specker group
On endomorphisms and automorphisms of some pure subgroups of the Baer-Specker group Open
The endomoprhism algebras of certain pure subgroups of the Baer-Specker group are investigated to answer a question of Irwin. Automorphism groups of these Abelian groups are investigated and a realization theme is established.
View article: Cotorsion-free algebras as endomorphism algebras in L - the discrete an topological cases.
Cotorsion-free algebras as endomorphism algebras in L - the discrete an topological cases. Open
The discrete algebras A over a commutative ring R which can be realised as the full endomorphism algebra of a torsion-free R-module have been investigated by Dugas and Gobel under the additional set-theoretic axiom of constructibility, V =…
View article: On Transitive and Fully Transitive Abelian P-Groups
On Transitive and Fully Transitive Abelian P-Groups Open
It is a long standing problem of A.L.S. Corner as to whether there exists a non-transitive fully transitive Abelian p-group with finite Ulm subgroups. The principal result of this paper shows that no such group exists for a special class o…
View article: Asymptotic methods in the spectral analysis of Sturm-Liouville operators
Asymptotic methods in the spectral analysis of Sturm-Liouville operators Open
,has been dubbed by the media as the ‘Celtic Tiger’ and certainly has been the focus of widespread international media (and indeed governmental) attention. Recall that Ireland, or at least the part excluding Northern Ireland, became an ind…
View article: On the socles of characteristically inert subgroups of Abelian <i>p</i>-groups
On the socles of characteristically inert subgroups of Abelian <i>p</i>-groups Open
We define the notion of a characteristically inert socle-regular Abelian p -group and explore such groups by focussing on their socles, thereby relating them to previously studied notions of socle-regularity. We show that large classes of …
View article: A Note On Clean Abelian Groups
A Note On Clean Abelian Groups Open
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 group…
View article: Some Transitivity Results For Torsion Abelian Groups
Some Transitivity Results For Torsion Abelian Groups Open
We introduce a new class of fully transitive and transitive Abelian p-groups and study the new concept of weak transitivity which is the missing link between full transitivity and transitivity.
View article: R-Hopfian and L-co-Hopfian Abelian Groups
R-Hopfian and L-co-Hopfian Abelian Groups Open
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,…
View article: On $p$-adic modules with isomorphic endomorphism algebras
On $p$-adic modules with isomorphic endomorphism algebras Open
We investigate pairs of modules over the ring of p-adic integers having isomorphic endomorphism algebras. In many cases this forces the modules to be isomorphic but there are two exceptional situations where isomorphism does not follow.
View article: On Maximal Relatively Divisible Submodules
On Maximal Relatively Divisible Submodules Open
In recent work Goebel and Goldsmith investigated the spectrum of maximal pure subgroups of certain Abelian groups. Here the situation relating to maximal pure submodules of a torsion-free module over an integral domain R is investigated. C…
View article: On torsion and mixed minimal Abelian groups
On torsion and mixed minimal Abelian groups Open
An abelian group is said to be minimal if it is isomorphic to all its subgroups of finite index. We obtain a complete characterisation of such groups in the torsion case; in the case of mixed groups of rank 1 we obtain a characterisation f…