A note on multiplicative property of the Smith normal form

In [2; p. 33]1 the following interesting result appears: If A and Bare n-square matrices over a principal ideal domain Rand g.c.d. (det (A), det (B»= 1 then S(AB)= S(A)S(B) where -SeA) is the Smith normal form of A. The purpose of this note is to present a simple proof of the result that uses elementary properties of compound matrices. LEMMA. Let Q=diag (q., ... , qn), P=diag (p., ... , Pn), qllqj' Pllpj,j=l, ... , nand g.c.d. (Pb qJ)= 1, i, j= 1, . .. , n. Let U be an n-square matrix with the property that g.c.d. (Ull' U2!, _ .. , Unl)= g.c.d. (Ull' U12, ... , Utn)= 1. Then the g.c.d. of all the entries in QUP is Plqt. PROOF. Obviously PlqlIQUP, i.e., Ptql divides every entry of QUP. Write QUP= PlqtD. Suppose that plD where p is a prime. It is simple to see that the first row and column of Dare respectively