Projective Quantum Modules and Projective Ideals of C *-algebras

We introduce in non-coordinate presentation the notions of a quantum algebra and of a quantum module over such an algebra. Then we give the definition of a projective quantum module and of a free quantum module, the latter as a particular case of the notion of a free object in a rigged category. (Here we say “quantum” instead of frequently used protean adjective “operator”). After this we discuss the general connection between projectivity and freeness.