Let $E/K$ be a field extension. It is a well known fact that all maximal subsets $A \subset E$ consisting of algebraically independent elements over $K$ have the same cardinality (which is by definition the transcendence degree of $E$ over $K$).
I suspect that this does not hold for arbitrary $K$-algebras $E$ (not even for finitely generated ones), but I don't know any counterexample. Can you name some preferably simple example where the statement doesn't hold?
Thank you in advance!
Let $E=K[X,Y,Z]/(XY,XZ)$. Then $x$ (the residue class of $X$) is a maximal algebraically independent subset, and $y,z$ are also algebraically independent.