Algebra of Matrix Arithmetic

We study the algebra of the arithmetic of integer matrices. A link is established between the divisor classes of matrices and lattices. The algebra of arithmetical functions of integral matrices is then shown to be isomorphic to an extension of the Hecke algebra, also called a Hall algebra in combinatorics. The dictionary helps translate results from one setting to another. One important application is the study of subgroups of a finite abelian group.