Compatibility in abstract Algebraic Structures

Compatible Hamiltonian pairs play a crucial role in the structure theory of integrable systems. In this paper we consider the question of how much of the structure given by compatibility is bound to the situation of hamiltonian dynamic systems and how much of that can be transferred to a complete abstract situation where the algebraic structures under consideration are given by bilinear maps on some module over a commutative ring. Under suitable modification of the corresponding definitions, it turns out that notions like, compatible, hereditary, invariance and Virasoro algebra may be transferred to the general abstract setup. Thus the same methods being so successful in the area of integrable systems, may be applied to generate suitable abelian algebras and hierarchies in very general algebraic structures.