RIMS - 1755 Monoid structure on square matrices over a PID By Kyoji SAITO July 2012 R ESEARCH I NSTITUTE FOR M ATHEMATICAL S CIENCES

We consider the set M(n,R)× of all square matrices of size n ∈ Z≥1 with non-zero determinants and coefficients in a principal ideal domain R. It forms a cancellative monoid with the matrix product. We develop an elementary theory of divisions by irreducible elements in M(n,R)×, and show that any finite set of irreducible elements of M(n,R)× has the right/left least common multiple up to a unit factor. As an application, we calculate the growth function PM(n,R)×,deg(t) and the skew growth function NM(n,R)×,deg(t) of the monoid M(n,R) ×. We get expressions PM(n,R)×,deg(exp(−s))= ζR(s)ζR(s−1)· · ·ζR(s−n+1) and NM(n,R)×,deg(exp(−s))= ∏ p∈{primes}(1−N(p))(1−N(p)) · · · (1−N(p)s−n+1), where ζR(s) is Dedekind zetafunction and N is the absolute norm on R. The structure of least common multiples in the monoid M(n,R)× studied above gives an elementary and direct proof of these decompositions, that is distinct from proofs by classical machinary.