The Extension Property in the Category of Direct Sum of Cyclic Torsion-Free Modules over a BFD

Abdelalim, Seddik; Chaïchaâ, Abdelhak; Garn, Mostafa El

doi:10.1007/978-3-030-35256-1_17

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2022

2023

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Cited by 3 publications

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“…In another work [2], the authors considered the category that contains abelian groups. They proved that if a module M is a direct sum of cyclic torsion-free modules over a BFD, the automorphisms of M that satisfy the property of the extension are none other than the homotheties of invertible ratio.…”

Section: Introductionmentioning

confidence: 99%

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The automorphisms having the extension property in a category of a finite direct sum of cyclic modules

Abdelalim¹,

Chaïchaâ²

2023

Discuss. Math. - General Algebra and Applications

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It is well known that the problem of characterizing the automorphisms, in the category of abelian groups, with the extension property is resolved [1]. But in other categories, it is a very difficult problem. This paper extends the result in [1] to a category of modules. Let A be a unique factorization integral domain (UFD). Consider M a direct finite sum of cyclic modules over A where Ann A (M ) = {0} and α an automorphism of M . We give a necessary and sufficient condition such that α satisfies the extension property.

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Section: Introductionmentioning

confidence: 99%

“…Certainly, one of the generalizations of [1] other than [2] is the case where all or some of the co-generators of M are not torsion-free. To do this, consider the following example which illustrates where the problem of the extension's ownership lies.…”

Section: Introductionmentioning

confidence: 99%