In classical algebraic number theory, there is no essential difference between fractional ideals and divisors. This fact is expressed in the fundamental theorem of arithmetic and its generalization to Dedekind domains. It states that the group of fractional ideals is free abelian and isomorphic to the divisor group. The paper gives a survey on noncommutative arithmetic, where the divisors still form a group, while the fractional ideals form a monoid which embeds into the divisor group. Ideal multiplication is extended to all divisors, which leads to a ring-like structure with a strong affinity to the theory of braces. The guiding principle of the survey is the surprising fact that in terms of divisors, multiplication of ideals is tantamount to the composition of functions.