The free module of rank over a nonzero unit ring , usually denoted , is the set of all -tuples (called vectors) where . We are particularly interested in , where is the ring of integers, which can be embedded into the vector space , where is the field of rationals. If a system of linear homogeneous equations with coefficients in has nontrivial solutions in rationals, it also has nontrivial solutions in integers. The unit vectors are interpreted as atoms, while all other vectors are molecules.