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In this paper, we introduce and investigate the latticial counterparts of the conditions (Ci), i = 1, 2, 3, 11, 12, for modules. In particular, we study the lattices satisfying the condition (C1), we call CC lattices (for Closed are Complements), i.e. the lattices such that any closed element is a complement, that are the latticial counterparts of CS modules (for Closed are Summands). Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory. [ABSTRACT FROM AUTHOR]

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]

This is the second part of the paper with the same title published inCommunications in Algebrain 2011. It contains applications of the Latticial Osofsky–Smith Theorem to Grothendieck categories and module categories equipped with a torsion theory. Various many different meanings spread in the literature of the relative concepts with respect to a hereditary torsion theory τ on Mod-Rlike τ-essential submodule, τ-complement submodule, τ-CS module, etc. are also discussed. [ABSTRACT FROM PUBLISHER]

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]

For an ordinal α, a modular lattice L with 0 and 1 is α-atomic if L has dual Krull dimension α but each interval [0,x] with x < 1 has dual Krull dimension <α. The properties of α-atomic lattices are presented and applied to module theory. The endomorphism ring of certain types of α-atomic modules is a local domain and hence there is a Krull–Schmidt type theorem for those α-atomic modules. [ABSTRACT FROM AUTHOR]