On quantum state of numbers

We introduce the notions of quantum characteristic and quantum flatness for arbitrary rings. More generally, we develop the theory of quantum integers in a ring and show that the hypothesis of quantum flatness together with positive quantum characteristic generalizes the usual notion of prime positive characteristic. We also explain how one can define quantum rational numbers in a ring and introduce the notion of twisted powers. These results play an important role in many different areas of mathematics and will also be quite useful in a subsequent work of the authors.