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Determinants used to be an important tool for the study of systems of linear equations. With the development of the abstract approach, based on the axioms of vector spaces, determinants lost their central role. Still, we feel that taking them out altogether from a text on linear algebra would be a mistake. Throughout the book we shall return to the use of special determinants for showing linear independence of vectors, invertibility of matrices, and coprimeness of polynomials. Since we shall need to compute determinants with polynomial entries, it is natural to develop the theory of determinants over a commutative ring with an identity.