Research ArticleNo Access

On a class of separable modules

Rachid Ech-chaouy

Department of Mathematics, Faculty of Sciences, Abdelmalek Essaâdi University, BP 2121 Tetouan, Morocco

E-mail Address: echaouirachid@yahoo.fr

Search for more papers by this author

Abdelouahab Idelhadj

Department of Mathematics, Faculty of Sciences, Abdelmalek Essaâdi University, BP 2121 Tetouan, Morocco

E-mail Address: idelhadj@gmail.com

Search for more papers by this author

, and

Rachid Tribak

Centre Régional des Métiers de l’Education, et de la Formation (CRMEF-TTH)-Tanger, Avenue My Abdelaziz, BP 3117 Souani, Tangier, Morocco

E-mail Address: tribak12@yahoo.com

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S1793557122500346Cited by:1 (Source: Crossref)

Abstract

A module $M$ is called $𝒜$ -separable if every proper finitely generated submodule of $M$ is contained in a proper finitely generated direct summand of $M$ . Indecomposable $𝒜$ -separable modules are shown to be exactly the simple modules. While direct summands of an $𝒜$ -separable module do not inherit the property, in general, the question of the stability under direct sums is unanswered. But we obtain some partial answers. It is shown that any infinite direct sum of $𝒜$ -separable modules is $𝒜$ -separable. Also, we prove that if $M_{1}$ and $M_{2}$ are $𝒜$ -separable modules such that $M_{1}$ is $M_{2}$ -projective, then $M_{1} \oplus M_{2}$ is $𝒜$ -separable. We conclude the paper by providing some characterizations of several classes of rings in terms of $𝒜$ -separable modules. Among others, we prove that the class of rings $R$ for which every (injective) $R$ -module is $𝒜$ -separable is exactly that of semisimple rings.

Communicated by C. C. Xi

Keywords:

AMSC: 16D10, 16D40, 16D80

References

1. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13 (Springer-Verlag, New York, 1974). Crossref, Google Scholar
2. R. Baer, Abelian groups without elements of finite order, Duke Math. J. 3(1) (1937) 68–122. Crossref, Google Scholar
3. J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules. Supplements and Projectivity in Module Theory, Frontiers in Mathematics (Birkhäuser, Basel, 2006). Google Scholar
4. A. Facchini, Module Theory. Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, (Birkhäuser, Switzerland, 1998). Crossref, Google Scholar
5. L. Fuchs, Infinite Abelian Groups, Vol. 36-I (Academic Press, London, 1970). Google Scholar
6. L. Fuchs, Infinite Abelian Groups, Vol. 36-II (Academic Press, London, 1973). Google Scholar
7. P. Griffith, Separability of torsion-free groups and a problem of J. H. C. Whitehead, Illinois J. Math. 12 (1968) 654–659. Crossref, Google Scholar
8. T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, Vol. 189 (Springer-Verlag, New York, 1999). Crossref, Google Scholar
9. W. Wm. McGovern, G. Puninski and P. Rothmaler, When every projective module is a direct sum of finitely generated modules, J. Algebra 315 (2007) 454–481. Crossref, Web of Science, Google Scholar
10. B. L. Osofsky, Rings all of whose finitely generated modules are injective, Pacific J. Math. 14 (1964) 645–650. Crossref, Web of Science, Google Scholar
11. B. L. Osofsky, A counter-example to a lemma of Skornjakov, Pacific J. Math. 15(3) (1965) 985–987. Crossref, Web of Science, Google Scholar
12. A. A. Tuganbaev, Rings Close to Regular, Mathematics and its Applications, Vol. 545 (Kluwer Academic Publishers, Dordrecht, 2002). Crossref, Google Scholar
13. R. Wisbauer, Foundations of Module and Ring Theory (Gordon and Breach Science Publishers, Philadelphia, 1991). Google Scholar
14. B. Zimmermann, Pure submodules of direct products of free modules, Math. Ann. 224 (1976) 233–245. Crossref, Web of Science, Google Scholar
15. H. Zöschinger, Quasi-separable und koseparable moduln über diskreten bewertungsringen, Math. Scand. 44 (1979) 17–36. Crossref, Google Scholar