  We prove for a large family of rings R that their lambda-pure global
dimension is greater than one for each infinite regular cardinal lambda. This
answers in negative a problem posed by Rosicky. The derived categories of such
rings then do not satisfy the Adams lambda-representability for morphisms for
any lambda. Equivalently, they are examples of well generated triangulated
categories whose lambda-abelianization in the sense of Neeman is not a full
functor for any lambda. In particular we show that given a compactly generated
triangulated category, one may not be able to find a Rosicky functor among the
lambda-abelianization functors.
