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Communicated by C.A. Weibel MSC: 16S90 16W30 16W60 06D99a b s t r a c t Let α and β be automorphisms on a ring R and δ : R → R an (α, β)-derivation. It is shown that if F is a right Gabriel filter on R then F is δ-invariant if it is both α and β-invariant. A consequence of this result is that every hereditary torsion theory on the category of right R-modules is differential in the sense of Bland (2006). This answers in the affirmative a question posed by Vaš (2007) and strengthens a result due to Golan (1981) on the extendability of a derivation map from a module to its module of quotients at a hereditary torsion theory.

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A Grothendieck category C is said to be locally finitely generated if the subobject lattice of every object in C is compactly generated, or equivalently, if C possesses a family of finitely generated generators. Every nonzero locally finitely generated Grothendieck category possesses simple objects. We shall call a Grothendieck category C indecomposable if C is not equivalent to a product of nonzero Grothendieck categories C 1 × C 2 . In this paper an example of an indecomposable nonlocally finitely generated Grothendieck category possessing simple objects is constructed, answering in the negative a sharper form of a question posed by Albu, Iosif, and Teply in [T. Albu, M. Iosif, M.L. Teply, Dual Krull dimension and quotient finite dimensionality, J. Algebra 284 (2005) 52-79].