The model-theoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their model-theoretic significance. It is these aspects that I will discuss in this article, although I will make some comments on the model theory of modules per se. Our default is that the term “module ” will mean (unital) right module over a ring (associative with 1) R. The category of such modules is denoted Mod-R, the full subcategory of finitely presented modules will be denoted mod-R, the

. Given a finite dimensional monomial algebra, one knows that some finite dimensional indecomposable modules may be described by words (finite sequences of letters) using as letters the arrows of the quiver and their formal inverses. To every word w, one can attach a so-called string module M(w). Here, we are going to construct certain infinite dimensional modules: We will consider N-words and Z-words (thus infinite sequences of letters) satisfying suitable periodicity conditions. To every such N-word or Z-word x, we describe an algebraically compact module C(x): This module C(x) is obtained from the corresponding string module M(x) as a kind of completion. Keywords. Algebraically compact modules, pure injective modules. Linearly compact modules. Quivers. Words, N-words, Z-words. String modules, Pr ufer modules, p-adic modules. Special biserial algebras. Kronecker modules. Shift endomorphism. 1. Finite Words and Finite Dimensional Modules Let k be a field. Let A be a finite dimensio...