Relativization, absolutization, and latticization in Ring and Module Theory

Abstract

In this survey paper we illustrate a general strategy which consists on putting a module-theoretical result into a latticial frame (we call it latticization), in order to translate that result to Grothendieck categories (we call it absolutization) and module categories equipped with hereditary torsion theories (we call it relativization). The renowned Hopkins–Levitzki Theorem and Osofsky–Smith Theorem from Ring and Module Theory, we will discuss in the last two sections of the paper, are among the most relevant illustrations of the power of this strategy.