  Let $T$ be an infinitely generated tilting module of projective dimension at
most one over an arbitrary associative ring $A$, and let $B$ be the
endomorphism ring of $T$. In this paper, we prove that if $T$ is good then
there exists a ring $C$, a homological ring epimorphism $B\ra C$ and a
recollement among the (unbounded) derived module categories $\D{C}$ of $C$,
$\D{B}$ of $B$, and $\D{A}$ of $A$. In particular, the kernel of the total left
derived functor $T\otimes_B^{\mathbb L}-$ is triangle equivalent to the derived
module category $\D{C}$. Conversely, if the functor $T\otimes_B^{\mathbb L}-$
admits a fully faithful left adjoint functor, then $T$ is a good tilting
module. We apply our result to tilting modules arising from ring epimorphisms,
and can then describe the rings $C$ as coproducts of two relevant rings.
Further, in case of commutative rings, we can weaken the condition of being
tilting modules, strengthen the rings $C$ as tensor products of two commutative
rings, and get similar recollements. Consequently, we can produce examples
(from commutative algebra and $p$-adic number theory, or Kronecker algebra) to
show that two different stratifications of the derived module category of a
ring by derived module categories of rings may have completely different
derived composition factors (even up to ordering and up to derived
equivalence),or different lengths. This shows that the Jordan-H\"older theorem
fails even for stratifications by derived module categories, and also answers
negatively an open problem by Angeleri-H\"ugel, K\"onig and Liu.

Representation Theory;Rings and Algebras