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.