In Miles Reid's Undergraduate Commutative Algebra, Exercise 1.3, we need to find counterexamples of lcm-gcd identity and modular law in the ring $A=k[X,Y]/(XY)$:
- $(I+J)(I\cap J)=IJ$;
- $I\cap(J+K)=(I\cap J)+(I\cap K)$.
for ideals $I,J\subseteq A$. In general, if $A$ isn't restricted to $k[X,Y]/(XY)$, we can consider $I=(X)$ and $J=(Y)$ in $A=k[X,Y]$. The left side of 1 should be $(X^2Y,XY^2)$ and the right side should be $(XY)$, so they're different. However, I want to understand the geometric meaning for the fallacy of the identity. When $A$ is a coordinate ring for an affine algebraic set, we cannot find the different zero sets of $(I+J)(I\cap J)$ and $IJ$ in general by Hilbert's Nullstellensatz. It seems that there's some infinitesimal information to distinguish.
I need some geometric interpretations for these, and then lead to an intuitive counterexample for 1,2. Any idea? Thanks!
EDIT: There's a counterexample: $I=(x),J=(x+y),K=(y)$, gratefully thanks to user26857.
