In this paper, we will show a reduction of ideal arithmetic, or more generally, of arithmetic of ZZ-modules of full rank in orders of number fields to problems of linear algebra over ZZ/mZZ, where m is a possibly composite integer. The problems of linear algebra over ZZ/mZZ will be solved directly, instead of either “reducing” them to problems of linear algebra over ZZ or factoring m and working modulo powers of primes and applying the Chinese Remainder theorem.