Hereditary arithmetics

The abstract arithmetic of non-commutative non-singular arithmetic curves (equivalently: the ideal theory of hereditary orders) is revisited in the framework of quantum B-algebras. It is shown that multiplication of ideals can be transformed into composition of functions. This yields a non-commutative “fundamental theorem of arithmetic” extending the classical one. Local hereditary arithmetics are presented by generators and relations and correlated with tubular quantum B-algebras. Main results are achieved by a divisor theory which furnishes the divisor group with a ring-like structure satisfying a 1-cocycle condition.