Abstract

Minimal approximations of modules,
or covers and envelopes of modules, were introduced as a tool
to approximate modules by classes of modules which are more
manageable. For a class C of R-modules, the aim is to
characterise the rings over which every module has a C-cover
or C-envelope. Moreover A-precovers and B-preenvelopes are
strongly related to the notion of a cotorsion pair (A,B).

In this talk we are interested in
the particular case that (P_1,B) is the cotorsion pair
generated by the modules of projective dimension at most one
(denoted P_1) over commutative rings. More precisely, we
investigate over which rings these cotorsion pairs admit
covers or envelopes. Furthermore, we interested in Enochs'
Conjecture in this setting, that is if P_1 is covering
necessarily implies that it is closed under direct limits. The
investigation of the cotorsion pair (P_1,B) splits into two
cases: when the cotorsion pair is of finite type and when it
is not. In this talk I will outline some results for the case
that the cotorsion pair is of finite type, where we consider
more generally a 1-tilting cotorsion pair over a commutative
ring.