TITLE: Showing that a certain map is not flat by explicit counterexample
QUESTION [5 upvotes]: I wish to show that the injection $k[y^2, y^3] \rightarrow k[y]$ is not flat. 

I know of geometric ways to see this, but I wish to see explicitly $k[y^2, y^3]$-modules (or localizations thereof) $0 \rightarrow M' \rightarrow M$ which does not remain injective upon tensoring with $k[y]$. Is such a search futile?

REPLY [6 votes]: I'd like to add a short algebraic proof (although this is not exactly what you asked for) which shows again the usefulness of Exercise 7.2 in Matsumura, Commutative Ring Theory: 

If $A\subset B$ are integral domains with the same field of fractions and $B$ is faithfully flat over $A$, then $A=B$.   

In your case $A=k[y^2,y^3]$ and $B=k[y]$. If the extension is flat, then it is faithfully flat (since it is integral) and therefore $A=B$, a contradiction.