  Let $A$ be a homological category and $U\colon B\to A$ be a faithful
conservative right adjoint. We introduce the notion of relative ideal with
respect to $U$, and we show that, under suitable conditions, any object of $A$
can be seen as a relative ideal of some object in $B$. We then develop a case
study. We first prove that the category of hoops is semi-abelian and that the
category of MV-algebras is protomodular, then we apply our results to the
forgetful functor from the category of MV-algebras to the category of Wajsberg
hoops.
