# Linear algebra: Modules over principal domains and similarity

## Abstract

This chapter treats further properties of modules over rings which are not fields. We have a discussion of the order of module elements not unlike the order of an element of a group or the characteristic of a ring. We resolve elements into sums of elements of elements of relatively prime orders. We introduce determinant divisors invariant under matrix equivalence. Using the theory developed for modules over principal ideal domains we find the invariant factor matrix: a canonical form for the equivalence of matrices over a principal domain. We solve linear equations with coefficients in a principal domain. We give a direct sum resolution of a finitely generated module over a principal domain. We consider the relation of similarity of matrices with entries in a field and apply the theorems of this chapter to yield several canonical forms. The technique is to construct, from a given vector space and endomorphism, a new module, resolve this module into a direct sum of cyclic submodules, and use this resolution to produce a basis for the vector space which yields for the endomorphism an especially simple matrix: the canonical form. We close with a study of the characteristic equation and characteristic values.

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