Throughout this chapter *R *is a principal ideal domain with unity. Thus, in particular, *R *may be a field. Recall that the rank of a finitely generated free *R*-module is defined to be the number of elements in any basis of the free module (Chapter 14, Section 5). For an *m *× *n *matrix *A *with entries in *R *we define the column rank and the row rank of *A *in terms of the rank of a certain free *R*-module, and we show in Section 3 that these two ranks are equal – the common value being known as the *rank o ƒ A. *But first we need a key lemma.

**1.1 Lemma.** *Let R be a principal ideal domain, and let F be a free R-module with a basis consisting of n elements. Then any submodule K of F is also free ...*

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