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## CHAPTER 20

### Smith normal form over a PID and rank

#### 1 Preliminaries

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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