## associated prime

What is the difference between an associated prime and an embedded prime?

Consider C[x,y]/<x2, xy>.

The primary decomposition of < x2, xy > is <x> $\cap$ <x,y>2, and is this also an associate & an embedded prime?

What is the geometry behind Z(<x2, xy>) in C2?

One last question: if a curve or a surface is intersecting itself, is this curve or surface called embedded curve/surface?