On some radicals and proper classes associated to simple modules
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.
D. Buchsbaum, A note on homology in categories, Ann. Math. 69(1) (1959) 66–74.
J. Clark, C. Lomp, N. Vanaja, R. Wisbauer, Lifting Modules, Frontiers in Mathematics, Birkhäuser, Basel, 2006.
N. V. Dung, D. V. Huynh, P. Smith, R. Wisbauer, Extending Modules, Pitman Research Notes in Mathematics, Harlow, Longman, 1994.
C. F. Preisser Montaño, Proper classes of short exact sequences and structure theory of modules, Ph.D. Thesis, Düsseldorf, 2010.
B. Stenström, Rings of Quotients, Springer, Berlin, Heidelberg, New York, 1975.
Y. Zhou, Generalizations of perfect, semiperfect and semiregular rings, Algebra Colloq. 7(3) (2000) 305–318.