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In this paper we introduce and investigate the latticial counterpart of the module-theoretical concept of preradical. Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories. [ABSTRACT FROM AUTHOR]

Abstract: A Grothendieck category is said to be locally finitely generated if the subobject lattice of every object in is compactly generated, or equivalently, if possesses a family of finitely generated generators. Every nonzero locally finitely generated Grothendieck category possesses simple objects. We shall call a Grothendieck category indecomposable if is not equivalent to a product of nonzero Grothendieck categories . 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]. [Copyright &y& Elsevier]

Abstract: In [R. Colpi, K.R. Fuller, Tilting objects in abelian categories and quasitilted rings, Trans. Amer. Math. Soc., in press] tilting objects in an arbitrary abelian category are introduced and are shown to yield a version of the classical tilting theorem between and the category of modules over their endomorphism rings. Moreover, it is shown that given any faithful torsion theory in , for a ring R, the corresponding Heart is an abelian category admitting a tilting object which yields a tilting theorem between the Heart and . In this paper we first prove that is a prototype for any abelian category admitting a tilting object which tilts to in . Then we study AB-type properties of the Heart and commutations with direct limits. This allows us to show, for instance, that any abelian category with a tilting object is AB4, and to find necessary and sufficient conditions which guarantee that is a Grothendieck or even a module category. As particular situations, we examine two main cases: when is hereditary cotilting, proving that is Grothendieck and when is tilting, proving that is a module category. [Copyright &y& Elsevier]