Simple-separable modules

Yıl 2024, Early Access, 1 - 22

Rachid Ech-chaouy Rachid Trıbak

https://doi.org/10.24330/ieja.1449459

Öz

A module $M$ over a ring is called simple-separable if every simple submodule of $M$ is contained in a finitely generated
direct summand of $M$. While a direct sum of any family of simple-separable modules is shown to be always simple-separable, we prove that
a direct summand of a simple-separable module does not inherit the property, in general. It is also shown that an injective module $M$
over a right noetherian ring is simple-separable if and only if $M=M_1 \oplus M_2$ such that $M_1$ is separable and $M_2$ has zero socle.
The structure of simple-separable abelian groups is completely described.

Anahtar Kelimeler

Separable module, simple-separable module, V-ring, $\pi$-V-ring

Kaynakça

• F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Grad. Texts in Math., 13, Springer-Verlag, New York-Heidelberg, 1974.
• R. Baer, Abelian groups without elements of finite order, Duke Math. J., 3(1) (1937), 68-122.
• W. H. Caldwell, Hypercyclic rings, Pacific J. Math., 24 (1968), 29-44.
• J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules. Supplements and Projectivity in Module Theory, Front. Math., Birkhauser, Verlag, Basel, 2006.
• R. Ech-chaouy, A. Idelhadj and R. Tribak, On coseparable and $\mathfrak{m}$-coseparable modules, J. Algebra Appl., 20(3) (2021), 2150031 (20 pp).
• R. Ech-chaouy, A. Idelhadj and R. Tribak, On a class of separable modules, Asian-Eur. J. Math., 15(2) (2022), 2250034 (17 pp).
• C. Faith, On Köthe rings, Math. Ann., 164 (1966), 207-212.
• L. Fuchs, Infinite Abelian Groups, vol. I, Pure Appl. Math., 36, Academic Press, New York-London, 1970.
• L. Fuchs, Infinite Abelian Groups, vol. II, Pure Appl. Math., 36, Academic Press, New York-London, 1973.
• P. Griffith, Separability of torsion free groups and a problem of J. H. C. Whitehead, Illinois J. Math., 12 (1968), 654-659.
• H. Harui, On injective modules, J. Math. Soc. Japan, 21 (1969), 574-583.
• Y. Hirano, On injective hulls of simple modules, J. Algebra, 225(1) (2000), 299-308.
• I. Kaplansky, Infinite Abelian Groups, University of Michigan Press, Ann Arbor, 1969.
• T. Y. Lam, Lectures on Modules and Rings, Grad. Texts in Math., 189, Springer-Verlag, New York, 1999.
• T. Y. Lam, Exercises in Modules and Rings, Probl. Books in Math., Springer, New York, 2007.
• W. Wm. McGovern, G. Puninski and P. Rothmaler, When every projective module is a direct sum of finitely generated modules, J. Algebra, 315(1) (2007), 454-481.
• S. H. Mohamed and B. J. Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Note Ser., 147, Cambridge University Press, Cambridge, 1990.
• B. J. Müller, On semi-perfect rings, Illinois J. Math., 14 (1970), 464-467.
• A. Ç. Özcan, A. Harmanci and P. F. Smith, Duo modules, Glasg. Math. J., 48(3) (2006), 533-545.
• M. Rayar, On small and cosmall modules, Acta Math. Acad. Sci. Hungar., 39(4) (1982), 389-392.
• A. Rosenberg and D. Zelinsky, Finiteness of the injective hull, Math. Z., 70 (1958/59/1959), 372-380.
• D. W. Sharpe and P. Vamos, Injective Modules, Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge University Press, London-New York, 1972.
• P. Vamos, The dual of the notion of " finitely generated", J. London Math. Soc., 43 (1968), 643-646.
• H. Zöschinger, Quasi-separable und koseparable moduln über diskreten bewertungsringen, Math. Scand., 44 (1979), 17-36.