I have been doing some reflecting and reading around the two issues/problems mentioned in my post above.
First problem/issue was as follows:
"My first problem with this example is as follows:
How can we demonstrate the the ideal (x,y) in k[x,y] is maximal"
In the excellent book "Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra" by David Cox, John Little and Donal O'Shea we find the following theorem (and its proof) on pages 201-202.
Proposition 9. If k is any field, an ideal of the form
Now (x, y) is of the form mentioned in Cox et al Proposition 9 since and so by Cox et al Proposition 9, (x,y) is maximal
Can someone confirm that this is correct.
Now reflecting on my second problem/issue.