The Baer-Kaplansky Theorem for a Class of Global Mixed Groups

Files, Steve T.; Wickless, W.

doi:10.1216/rmjm/1181072075

Cited by 14 publications

(1 citation statement)

References 15 publications

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“…Consequently, results of this form are often referred to in the literature as 'Baer-Kaplansky Theorems'-see, for example, [3,12] or [14] for a small selection of such results. Subsequently Hauptfleisch [10] and Wolfson [16] extended the result to certain classes of torsion-free Abelian groups and modules; in particular Wolfson showed that if G, H are reduced torsion-free R-modules then E R (G) ∼ = E R (H) if and only if G ∼ = H. The situation for mixed modules and Abelian groups is difficult but some interesting results were obtained in [15] and [4]. Recently the authors, working on an unpublished idea of Corner, have considered the corresponding problem where isomorphism of the endomorphism algebras is replaced by anti-isomorphism, [9].…”

Section: Introductionmentioning

confidence: 99%

On $p$-adic modules with isomorphic endomorphism algebras

Goldsmith¹,

White²

2021

Irish Math. Soc. Bull.

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Section: Introductionmentioning

confidence: 99%

On $p$-adic modules with isomorphic endomorphism algebras

Goldsmith¹,

White²

2021

Irish Math. Soc. Bull.

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Direct Sums of Self-Small Mixed Groups

Files

Wickless

1999

Journal of Algebra

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The class G G of self-small mixed abelian groups of finite rank has recently been the focus of a number of investigations. It is known that, up to quasi-isomorphism, G G is dual to the category of locally free torsion-free abelian groups of finite rank. We utilize the reduced mixed groups in G G as building blocks of a more extensive class Ý G G, the smallest class containing the reduced groups in G G that is closed under taking infinite direct sums and direct summands. Our central result is the determination of a complete set of isomorphism invariants for the groups in Ý G G. We supplement this broad classification theorem with an investigation of the fine structure of completely decomposable groups in Ý G G. ᮊ 1999 Academic Press CANCELLATION AND THE CLASS G GA ring R has one in the stable range if, whenever the equation r s q 1 1 r s s 1 holds in R, there exists an element s g R such that r q r s is a 2 2 1 2 w x unit of R. In B , Bass proves that if R factored by its Jacobson radical is Artinian, then one is in the stable range of R. Plainly then, finite rings and finite dimensional rational algebras have one in the stable range.All groups considered will be additive abelian groups and all homomor-Ž . phisms will be Z-homomorphisms.

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Some mixed abelian groups as modules over the ring of pseudo—rational numbers

Fomin¹

1999

Abelian Groups and Modules

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The Baer-Kaplansky Theorem for a Class of Global Mixed Groups

Cited by 14 publications

References 15 publications

On $p$-adic modules with isomorphic endomorphism algebras

On $p$-adic modules with isomorphic endomorphism algebras

Direct Sums of Self-Small Mixed Groups

Some mixed abelian groups as modules over the ring of pseudo—rational numbers

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