  It is well-known that a class of all modules, which are torsion-free with
respect to a set of ideals, is closed under injective envelopes. In this paper,
we consider a kind of a dual to this statement - are the divisibility classes
closed under flat covers? - and argue that this is seldom the case. More
precisely, we show that the class of all divisible modules over an integral
domain R is closed under flat covers if and only if R is almost perfect. Also,
we show that if the class of all s-divisible modules, where s is a regular
element of a commutative ring R, is closed under flat covers then the quotient
ring R/sR satisfies some rather restrictive properties. The question is
motivated by the recent classification [11] of tilting classes over commutative
rings.
