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.
Highlights
For an associative ring R with identity, let M be a unitary right R-module
For a unitary right module M, there are two known partitions of simple modules in the category σ[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
Summary
For an associative ring R with identity, let M be a unitary right R-module. the category σ[M ] is a Grothendieck category which consists of all right R-modules subgenerated by M , that is, submodules of M -generated right R-modules. The first one partitions them into the pair of classes (S(I ), S(M )), where S(I ) is the class of simple M -injective modules in σ[M ] and S(M ) is the class of simple M -small modules in σ[M ] [4, 4.2]. The second one partitions them into the pair of classes (S(P), S(S )), where S(P) is the class of simple M -projective modules in σ[M ] and S(S ) is the class of simple M -singular modules in σ[M ] [3, 8.2]. We are interested in such characterizations for possible inclusions between the four classes of simple modules S(I ), S(M ), S(P) and S(S ). They are inspired by and complete results by Preisser Montaño [5]
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