In this document I describe how the standard way the socle operator is set up for a module, and a neater way is can be set up using the lattice of submodules. I indicate why this way is neater by showing how the construction can be iterated. I go through the various constructions first for the straight socle, and then for the socle relative to a given hereditary torsion theory.

Let τ be a hereditary torsion theory on the category R-Mod of left R-modules over an associative unitary ring R. We introduce the notion of τ-natural class as a class of modules closed under τ-dense submodules, direct sums, and τ-injective hulls. We study connections between certain conditions involving τ-(quasi-)injectivity in the context of τnatural classes, generalizing results established b...

We characterize the hereditary torsion pairs of finite type in the functor category of a ring R associated to tilting torsion pairs in the category of R-modules. Moreover, we determine a conditions under which they give rise to TTF triples.

To study the set of torsion classes a finite dimensional basic algebra over field, we use decomposition, called sign-decomposition, parameterized by elements {±1}n where n is number simple modules. If A an with radical square zero, then for each ϵ∈{±1}n there hereditary Aϵ! zero and bijection between associated to ϵ faithful Aϵ!. Furthermore, this preserves property being functorially finite. F...