I'm studying commutative algebra and I need to find, for the course I'm attending, the primary decompositions of the following ideals, with some extra questions:
In $\mathbb{C}[x,y]$, find, if you can, two primary decompositions of the ideal $(x^2y, xy^2)$, and find all isolated primes. Is the ideal radical? Is the decomposition unique? In the same ring, find a primary decomposition of $H = (y - x^2, x - y)$, and find whether it's radical or not.
In $\mathbb{C}[x,y,z]$, find two different primary decompositions of $I = (xyz^2,xy^2)$, and the associated isolated primes. In the same ring, find a primary decomposition of $J = (x,z,x^2 + y^2 - 1)$.
Can somebody help me? We didn't really do Grobner bases that much, we just defined them in general, so a different approach would be appreciated. Also, I have literally no clue where to start, I couldn't follow the course and in the notes there's no mention of an example whatsoever.
Thanks in advance to whoever helps me.