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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.

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The $A$-module $A$ is clearly free, because it has basis $\langle1\rangle.$ However, it can also be generated over $A$ by $x$ and $1-x.$ For example, we have $x+ (1-x)=1,$ so any $f\in A=M$ can be expressed as $f\cdot x +f\cdot (1-x)$ (where the $f$ here are "scalars").

However, notice that $\langle x,1-x\rangle$ is not an $A$-basis, since $(1-x)\cdot x + (-x)\cdot (1-x)=0.$ On the other hand, neither of $x$ or $1-x$ would suffice on its own to generate $A$; the degree of $f\cdot x$ and $f\cdot (1-x)$ is always $\deg(f)+1$ for any "scalar" $f.$ This is what he means by "irredundant."