I'll survey joint works with Gena Puninski (Minsk) on the model theory of modules over a Bézout domain \(B\).
I'll discuss width and existence of superdecomposable pure injective modules. I'll also deal with decidability when \(B\) is obtained by the so called D+M construction
(namely \(B = D + X Q[X]\) where \(D\) is a principal
ideal domain and \(Q\) is its field of fractions).