Semiprime and weakly compressible modules

Year 2016, Volume: 45 Issue: 2, 343 - 353, 01.04.2016

N. Dehghani M. R. Vedadi

Abstract

An R-module M is called semiprime (resp. weakly compressible) if it is
cogenerated by each of its essential submodules (resp. HomR(M, N)N
is nonzero for every 0 6= N ≤ MR). We carry out a study of weakly
compressible (semiprime) modules and show that there exist semiprime
modules which are not weakly compressible. Weakly compressible modules with enough critical submodules are characterized in different ways.
For certain rings R, including prime hereditary Noetherian rings, it
is proved that MR is weakly compressible (resp. semiprime) if and
only if M ∈ Cog(Soc(M) ⊕ R) and M/Soc(M) ∈ Cog(R) (resp. M ∈
Cog(Soc(M) ⊕ R)). These considerations settle two questions, namely
Qu 1, and Qu 2, in [6, p 92].

Keywords

Krull dimension, semiprime module, singular semi-Artinian ring, weakly compressible module

References

• Avraamova, O. D. A generalized density theorem, in Abelian Groups and Modules, Tomsk Univ. Publ. 8, 3-16, 1989.
• Bican, L., Jambor, P., Kepka, T. and N¡emec, P. Prime and coprime modules, Fund. Math. 57, 33-45, 1980.
• Haghany, A. and Vedadi, M. R. Study of semi-projective retractable modules, Algebra Colloquium 14 (3), 489-496, 2007.
• Jira´sko, J. Notes on generalized prime and coprime modules, I, Comm. Math. Univ. Carolinae 22 (3), 467-482, 1981.
• Lam, T. Y. Lectures on Modules and Rings, Grad. Texts in Math. 139, Springer, New York, 1998.
• Lomp, C. Prime elements in partially ordered groupoids applied to modules and Hopf algebra actions, J. Algebra Appl. 4 (1) 77-97, 2005.
• McConnell, J. C. and Robson, J. C. Non-commutative Noetherian Rings, Wiley Interscience, New York, 1987.
• Samsonova, I. V. Weakly compressible abelian groups, Comm. Moscow. Math. Soc. 187-188, 1993.
• Smith, P. F. and Vedadi, M. R. Submodules of direct sums of compressible modules, Comm. Algebra. 36, 3042-3049, 2008
• Toloei, Y. and Vedadi, M. R. On rings whose modules have nonzero homomorphisms to nonzero submodules, Publ. Mat. 57 (1), 107-122, 2012.
• Wisbauer, R. Modules and Algebras: Bimodule Structure and Group Action on Algebras, Pitman Monograhs 81, Addison-Wesley Longman, 1996.
• Zelmanowitz, J. Weakly semisimple modules and Density theory, Comm. Algebra 21 (5), 1785-1808, 1993.