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• W21110596 abstract "Let R be an associative ring with identity and κ, an infinite regular cardinal. A left R-module M is said to be κ ≤-generated (resp. κ <-generated) if there exist a generator set {m i } I of M of cardinality at most κ (resp. strictly smaller than κ And a module M is called κ-separable if any subset X of M of cardinality strictly smaller than κ is contained in a κ <-generated direct summand of M. Let us no that any direct sum of κ <-generated modules is clearly a κ-separable module that we will call trivial. This notion of separability can be extended as follows. Given an infinite regular cardinal κ and a non-empty class of modules ( mathcal{C} ), we may say that a module M is (κ, ( mathcal{C} ))-separable if each subset X of M of cardinality strictly smaller than κ is contained in a direct summand of M belonging to ( mathcal{C} ). Again, we will say that the (κ, ( mathcal{C} ))-separable module M is non-trivial if it is not a direct sum of elements in ( mathcal{C} ). These modules present a pathological behavior in the sense that they have enough direct summands belonging to ( mathcal{C} ), but they are not direct sums of modules in ( mathcal{C} )." @default.
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• W21110596 date "2008-08-21" @default.
• W21110596 modified "2023-09-27" @default.
• W21110596 title "On the Construction of Separable Modules" @default.
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• W21110596 doi "https://doi.org/10.1007/978-3-7643-8742-6_14" @default.
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