Relative ideals in homological categories with an application to MV-algebras

Let A be a homological category and U: B → A be a faithful conservative right adjoint. We introduce the notion of relative ideal with respect to U and 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.