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We develop an elementary theory of divisibility on the monoid $M(n,R)^\times$consisting of all square matrices of size $n\ge 1$ of non-zero determinantswith coefficients in a principal ideal domain $R$. In particular, we show thatany finite subset of the monoid has the least left common multiple up to aright unit factor. When $R$ is residue finite, we consider a signed generatingseries, called the skew growth function, of least common multiples of finiteright equivalence classes of irreducible elements. As an elementary applicationof the divisibility theory, we show that the skew-growth function decomposesinto Euler products.

The skew-growth function on the monoid of square matrices