Publisher Summary 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.

On linear algebra over commutative rings

The structure of the complex RHomR(R/I,R) is explored for an Ulrich ideal I in a Cohen–Macaulay local ring R. As a consequence, it is proved that in a onedimensional almost Gorenstein but non-Gorenstein local ring, the only possible Ulrich ideal is the maximal ideal. It is also studied when Ulrich ideals have the same minimal number of generators.

Ulrich ideals and almost Gorenstein rings

The structure of the complex $\operatorname{\mathbf{R}Hom}_R(R/I,R)$ is explored for an Ulrich ideal $I$ in a Cohen-Macaulay local ring $R$. As a consequence, it is proved that in a one-dimensional almost Gorenstein but non-Gorenstein local ring, the only possible Ulrich ideal is the maximal ideal. It is also studied when Ulrich ideals have the same minimal number of generators.

This chapter discusses ideals and modules. The theory of ideals was created by Dedekind from the theory of the ideal numbers. Modules are created by using the axioms of vector spaces but allowing the scalars to be in any ring. Modules appear quite naturally and are commutative. The notion of modules is also important in representation theory. The extension to modules of natural notions or properties of ideals is called “modulation.” In addition, there exist very useful characterizations of rings by the properties of some types of modules over them, such that a ring R is a field if every module over it is free. The most important notion is the notion of ideals. A diagram of morphisms is commutative if the composite of morphisms from one module to another does not depend on the path taken. The inverse limit is connected with the notion of intersection and the direct limit, with that of union.

Ideals and Modules

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Ideals and Modules