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Abstract Algebra American Mathematical Society Assume columns of WK columns preceding common left multiple commutative companion matrix congruence Corollary 1.8 defined degree kn degree less degree Sy direct product divisor d(X divisor of Q elements inj exist a unimodular factor family of right greatest common divisor Hence Hermite form Introduction to Abstract kn unknowns leading coefficient unity least common left left associates linearly dependent M. H. Ingraham main diagonal element matrices with elements maximum degree modular modulo necessary and sufficient non-commutative non-singular non-zero obtained occurs parameter parametric family polynomial principal ideal ring proper of degree proper right divisors relatively prime rows square of order sufficient condition t-th Theorem 1.4 Therefore,there thesis triangular form unilateral equation unimodular matrix unique solution uniquely determined University of Wisconsin zero