I'm reading about primary decomposition from Atiyah and Macdonald's book. One of my friend told me that it is possible to find Associated primes and Primary decomposition geometrically. I tried to found the geometric way in Atiyah and Macdonald's book, but it is not explained in the book. Can someone explain for the following example:
Let $I=(x^2+xy,y^2-1) \subset k[x,y]$. Find Associated primes and Primary decomposition geometrically.
I don't know how to find the associated primes and primary decompositions geometrically, but this particular example is easy enough since $I$ is a radical ideal (why?) and therefore it is the intersection of its minimal prime ideals, that is, $$I=(X,Y-1)\cap(X,Y+1)\cap(X+1,Y-1)\cap(X-1,Y+1).$$