Abstract
This chapter reviews the results and methods in linear algebra over commutative rings. This theory is formed by systems of linear equations and linear dynamical systems over commutative rings. The results on the characterization of some classes of rings in terms of systems of linear equations are discussed. An extended version to commutative rings of Brunovsky's theorem, the classification of some special linear systems over a principal ideal domain or over a local ring and the characterization of the point wise feedback relation is described. The dynamical feedback classification problem is also provided. The stabilization problem in classical control theory is the origin of the pole-shifting problem. Different classes of rings appear in the study of pole shifting over commutative rings. The characterization of these rings in algebraic terms is discussed.
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