Tag Archives: Fitting ideals

Linear algebra over rings

Test your intuition: is the following true or false?

Assertion 1: If A is a square matrix over a commutative ring R, the rows of A are linearly independent over R if and only if the columns of A are linearly independent over R.

(All rings in this post will be nonzero commutative rings with identity.)

And how about the following generalization?

Assertion 2: If A is an m \times n matrix over a commutative ring R, the row rank of A (the maximum number of R-linearly independent rows) equals the column rank of A (the maximum number of R-linearly independent columns).

If you want to know the answers, read on…

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Fitting ideals of modules

In my previous post, I presented a proof of the existence portion of the structure theorem for finitely generated modules over a PID based on the Smith Normal Form of a matrix. In this post, I’d like to explain how the uniqueness portion of that theorem is actually a special case of a more general result, called Fitting’s Lemma, which holds for arbitrary commutative rings.

Hans Fitting

We begin by proving that one can characterize the diagonal entries in the Smith Normal Form of a matrix A over a PID in an intrinsic way by relating them to the GCD of the k \times k minors of A for all k. Actually, since the GCD isn’t defined for general rings, we will instead consider the ideal generated by the k \times k minors (which makes sense for any ring, and is the ideal generated by the GCD in the case of a PID).

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