Test your intuition: is the following true or false?

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

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

And how about the following generalization?

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

If you want to know the answers, read on…