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In this paper, which is a continuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions ( C) in modular lattices, and applications, J. Algebra Appl. 15 (2016), http: dx.doi.org/10.1142/S0219498816500018], we investigate the latticial counterparts of some results about modules satisfying the conditions ( C) or ( C). Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories. [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]