Prove that C[x,y]⟨x2−y2−1⟩ is an integral domain such that all the nonzero prime ideals are maximal.

manpreet Best Answer 2 years ago

Let R=C[x,y]⟨x2−y2−com/tag/1">1⟩R=C[x,y]⟨x2−y2−com/tag/1">1⟩.

Prove that

(1) RR is an integral domain.
(2) Any nonzero prime ideal of RR is maximal.

My idea:

(1) The first part is easy. Since x2−y2−com/tag/1">1∈C[y][x]x2−y2−com/tag/1">1∈C[y][x] is Eisenstein with respect to the prime y+iy+i, it is irreducible. Hence the ideal generated by it is a prime ideal.

(2) I am struggling with this part. I have one approach (not sure if correct).

Approach : Let I=⟨x2−y2−com/tag/1">1⟩I=⟨x2−y2−com/tag/1">1⟩. Then R=C[x,y]/IR=C[x,y]/I. Any ideal of RR is of the form J/I