We develop the theory of trace modules up to isomorphism and explore relationship between preenveloping classes property being a module, guided by question whether given module is in preenvelope. As consequence we identify new examples ideals modules, characterize several rings with focus on Gorenstein regular properties.

Let R be a ring with identity and C(R) denote the category of complexes of R-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp. Gorenstein injective) modules. We show that the homotopy category of projective complexes over R, denoted K(Prj C(R)), is always well generated and is compactly generated p...