quantum determinants in ribbon category

The aim of this paper is to introduce an abstract notion of determinant which we call quantum determinant, verifying the properties of the classical one. we introduce r−basis and r−solution on rigid objects of a monoidal 𝐴𝑏−category, for a compatibility relation r, such that we require the notion of duality introduced by joyal and street, the notion given by yetter and freyd and the classical one, then we show that r−solutions over a semisimple ribbon 𝐴𝑏−category form as well a semisimple ribbon 𝐴𝑏−category. this allows us to define a concept of so-called quantum determinant in ribbon category. moreover, we establish relations between these and the classical determinants. some properties of the quantum determinants are exhibited.