Let R be a principal ideal domain. For each positive integer rand sequence of nonzero ideals I₁ ⊃ I₂ ⊃ • . • ⊃ I_r choose a sequence d₁, ... , dr ϵ R such that (d_j) = I_j and d₁| d₂ I· · · I d_r. For a given pair of positive integers (n, m), let S be these to fall n X m matrices of the for m where r = 1, 2, ... ,min(n, m) and L, is an r X r diagonal matrix with main diagonal one of the chosen sequences d₁, . .. , d_r. Show that S is a set of canonical forms under equivalence for the set of all n X m matrices over R.

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LT 8). 0