# 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).

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).