DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

Cite Score (2023): 0.6

SJR (2023): 0.214

SNIP (2023): 0.604

Index Copernicus (2022): 121.02

H-Index: 5

Discussiones Mathematicae - General Algebra and Applications

Article in volume

Authors:

S. Abdelalim

Seddik Abdelalim

Laboratory of Topology
Algebra, Geometry and Discrete Mathematics
Departement of Mathematical and Computer Sciences
Faculty of Sciences Ain Chock Hassan II
University of Casablanca BP 5366 Maarif, Casablanca, Morocco

email: seddikabd@hotmail.com

A. Chaichaa

Abdelhak Chaichaa

Laboratory of Topology
Algebra, Geometry and Discrete Mathematics
Departement of Mathematical and Computer Sciences
Faculty of Sciences Ain Chock Hassan II
University of Casablanca BP 5366 Maarif, Casablanca, Morocco

email: abdelchaichaa@gmail.com

M. El garn

Mostafa El garn

Laboratory of Topology
Algebra, Geometry and Discrete Mathematics
Departement of Mathematical and Computer Sciences
Faculty of Sciences Ain Chock Hassan II
University of Casablanca BP 5366 Maarif, Casablanca, Morocco

email: elgarnmostafa@gmail.com

0000-0003-0419-7560

Title:

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

PDF

Source:

Discussiones Mathematicae - General Algebra and Applications 43(1) (2023) 111-120

Received: 2020-06-08 , Revised: 2021-10-05 , Accepted: 2021-10-13 , Available online: 2023-01-11 , https://doi.org/10.7151/dmgaa.1411

Abstract:

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 $\alpha$ an automorphism of $M$. We give a necessary and sufficient condition such that $\alpha$ satisfies the extension property.

Keywords:

integral domain, factorization, module, automorphism, torsion and torsion-free

References:

  1. S. Abdelalim and H. Essannouni, Characterization of the automorphisms of an Abelian group having the extension property, Portugaliae Math. (Nova) 59 (3) (2002) 325–333.
    https://doi.org/org/10.1016/j.jtusci.2015.02.009
  2. S. Abdelalim, A. Chaicha and M. El garn, The Extension Property in the Category of Direct Sum of Cyclic Torsion-Free Modules over a BFD, in: The Moroccan Andalusian Meeting on Algebras and their Applications 2018, Ap, Springer, Cham. (Ed(s)), (2020) 313–323.
    https://doi.org/org/10.1007/978-3-030-35256-1-17
  3. M. Barry, These: Caractérisation des anneaux commutatifs pour lesquels les modules vérifant (I) sont de types finis, Dissertation Doctorale (Université Cheikh anta diop de Dakar, Faculté des Sciences et Techniques, Département de Mathématiques et Informatique, 1998) 22–22.
  4. L. Ben Yakoub, Sur un Théorème de Schupp, Portugaliae Math. (Nova) 51 (2) (1994) 231–233.
    https://doi.org/eudml.org/doc/47203
  5. E. Kunz, Introduction to commutative algebra and algebraic geometry (Springer Science & Business Media, 2013).
    https://doi.org/10.1007/978-1-4614-5987-3
  6. M.R. Pettet, On Inner Automorphisms of Finite Groups, Proceedings of the American Mathematical Society 106 (1) (1989) 87–90.
    https://doi.org/org/10.1090/S0002-9939-1989-0968625-8
  7. P.E. Schupp, A characterizing of inner automorphisms, in: Proc. Amer. Math. Soc. 101 (2) (1987) 226–228.
    https://doi.org/10.1090/S0002-9939-1987-0902532-X
Close