I would appreciate help (self-studier) with an example in Ried's "Undergrad. Commutative Algebra" p.41 in the chapter on modules.
"Even when $M$ is free, an irredundant family of generators is not necessarily a basis; for example $A$ = $k[X]$, $M = A$ and $(X, 1 - X)$."
To begin with I don't know how to read the last phrase: $M = A$ and $(X, 1 - X)$
Understanding that may be a big help, but I might yet be stuck. I do know what the math terms mean. I guess I want to produce a basis of $M$ - need clarification of what that is - to show $M$ is free. Then I need a family of generators, spanning $M$, and show that they are linearly dependent.
I would like to try this once I understand that last phrase as I mentioned. But in case I'm still stuck, I would appreciate any hints that I can subsequently look at.
Thanks very much.