In classical algebraic geometry one thinks about commutative algebras as algebras of functions on spaces. If the underlying space is also a group, the corresponding algebra of functions becomes a coalgebra. Both coalgebra and algebra structures are compatible with each other in the sense that the coproduct and counit are algebra maps. This example motivates studies of coalgebras with a compatible commutative algebra structure. Allowing further generalisations, one considers coalgebras with noncommutative algebraic structures, initially over fields, but eventually over commutative rings. Such algebras with compatible coalgebra structures are known as bialgebras and Hopf algebras and often are referred to as quantum groups.

There are numerous textbooks and monographs on bialgebras, Hopf algebras and quantum groups (the latter mainly addressed to a physics audience), in particular, classic texts by Sweedler [45] or Abe [1], or more recent works (Montgomery [37], Dǎscǎlescu, Nǎstǎsescu and Raianu [14]), including the ones motivated by the quantum group theory (e.g., Lusztig [30], Majid [33, 34], Chari and Pressley [11], Shnider and Sternberg [43], Kassel [25], Klimyk and Schmüdgen [26], Brown and Goodearl [7]). In the majority of these texts it is assumed at the beginning that all algebras and coalgebras are defined over a field. By making this assumption, the authors are excused from not considering some of the module-theoretic aspects of the discussed objects.