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Abelian group algebraic domain algebraic Integers algebraic number class number coefficients column class column from right common Ideal divisor common right divisor commutative condition that matrices corollary corresponding critical minor defined diagonal block diagonal element elements of H enlarged matrices example exist matrices exists a matrix exists a unimodular factors field finite number form H greatest common divisor greatest common right H is unique hence Hermite form implies invariant factors irreducible basis Irreducible polynomial k-by-k blocks k-by-k matrices kn-by-kn left associates LEFT-ASSOCIATED MATRICES Lemma 13 linear combination main diagonal matrices of rank matrices with elements matrix H matrix of 7t metric representative minimal basis minor determinants modulo the diagonal necessary and sufficient non-principal class non-principal ideal non-singular matrices non-zero rows polynomial f(x prime Ideals principal ideal ring problem proof reduced rows and columns Steinits sufficient condition Theorem theory thesis thst unimodular matrix zero rows