On some radicals and proper classes associated to simple modules

Year 2017, Volume: 4 Issue: 2 (Special Issue: Noncommutative rings and their applications), 123 - 129, 10.01.2017

Septimiu Crivei , Derya Keskin Tütüncü

https://doi.org/10.13069/jacodesmath.284943

Abstract

For a unitary right module $M$, there are two known partitions of
simple modules in the category $\sigma[M]$: the first one divides
them into $M$-injective modules and $M$-small modules, while the
second one divides them into $M$-projective modules and
$M$-singular modules. We study inclusions between the first two
and the last two classes of simple modules in terms of some
associated radicals and proper classes.

Keywords

Radical, Proper class, Simple module, Relative supplement submodule, M-injective module, M-projective module, M-small module, M-singular module

References

• [1] K. Al–Takhman, C. Lomp, R. Wisbauer, $\tau-$complemented and $\tau-$supplemented modules, Algebra Discrete Math. 3 (2006) 1–16.
• [2] D. Buchsbaum, A note on homology in categories, Ann. Math. 69(1) (1959) 66–74.
• [3] J. Clark, C. Lomp, N. Vanaja, R. Wisbauer, Lifting Modules, Frontiers in Mathematics, Birkhäuser, Basel, 2006.
• [4] N. V. Dung, D. V. Huynh, P. Smith, R. Wisbauer, Extending Modules, Pitman Research Notes in Mathematics, Harlow, Longman, 1994.
• [5] C. F. Preisser Montaño, Proper classes of short exact sequences and structure theory of modules, Ph.D. Thesis, Düsseldorf, 2010.
• [6] B. Stenström, Rings of Quotients, Springer, Berlin, Heidelberg, New York, 1975.
• [7] Y. Zhou, Generalizations of perfect, semiperfect and semiregular rings, Algebra Colloq. 7(3) (2000) 305–318.