query: If $I$ is not finitely generated and every ideal properly containing $I$ is finitely generated, then $I$ is prime?
positive_passages.docid: 6699
positive_passages.text: An ideal that is maximal among non-finitely generated ideals is prime.
negative_passages.docid: 87780
negative_passages.text: Is the ideal $(2,X+1)\subset\Bbb{Z}[X]$ prime, maximal or neither?
negative_passages.docid: 15904
negative_passages.text: Prime ideal and nilpotent elements
negative_passages.docid: 53469
negative_passages.text: C(0, 1) has infinitely many prime ideals not of the form Mp
negative_passages.docid: 53770
negative_passages.text: Let $R$ be a finite commutative ring. Show that an ideal is maximal if and only if it is prime.
negative_passages.docid: 147871
negative_passages.text: How do the prime ideals of $\mathbb{Z}_{k}$ look like?
negative_passages.docid: 151786
negative_passages.text: Proving that there is no ideal $I$ in $\mathbb{Z}[t]$ such that $\mathbb{Z}[t]/I \cong \mathbb{Q}$
negative_passages.docid: 15137
negative_passages.text: Does $IJ=IK\implies J=K$ always hold for integral domain and finitely generated nonzero ideal $I$?
negative_passages.docid: 111114
negative_passages.text: Prove that an ideal in $k[x,y]$ is a prime ideal
negative_passages.docid: 67669
negative_passages.text: Prime ideals of a finite direct product ring
negative_passages.docid: 93658
negative_passages.text: Proving that an ideal in a PID is maximal if and only if it is generated by an irreducible
negative_passages.docid: 155745
negative_passages.text: Ideals of $\mathbb{Z}[X]$
negative_passages.docid: 23099
negative_passages.text: Verifying that the ideal $(x^3-y^2)$ is prime
negative_passages.docid: 69710
negative_passages.text: Is it true that an ideal is primary iff its radical is prime?
negative_passages.docid: 344
negative_passages.text: Prove that if $p\in \mathbb{Z}$ is irreducible, then it is also prime.
negative_passages.docid: 103518
negative_passages.text: If $J$ is the ideal generated by all idempotents in a prime ideal, then $R/J$ has only trivial idempotents
negative_passages.docid: 90889
negative_passages.text: Prime ideals in a finite direct product of rings