query: Show that an integral domain with every strictly decreasing chain of ideals $I_1 \supset I_2\supset \cdots $ finite in length is a field.
positive_passages.docid: 23242
positive_passages.text: Commutative integral domain with d.c.c. is a field
negative_passages.docid: 7011
negative_passages.text: Integral domain with fraction field equal to $\mathbb{R}$
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: 150239
negative_passages.text: In an integral domain every non-zero prime ideal is invertible implies every non-zero proper ideal is invertible.
negative_passages.docid: 104486
negative_passages.text: Ideal in $\mathbb Z[x]$ which is not two-generated
negative_passages.docid: 62951
negative_passages.text: Is there a more direct way of proving that this ring is an integral domain?
negative_passages.docid: 14055
negative_passages.text: How to show that a finite commutative ring without zero divisors is a field?
negative_passages.docid: 93621
negative_passages.text: Prove that $\mathbb{Z}[\omega]$ is an integral domain
negative_passages.docid: 157585
negative_passages.text: Structure theorem of finite rings
negative_passages.docid: 99423
negative_passages.text: Is it true every ideal of $\mathbb{Z}[x]$ is finitely generated?
negative_passages.docid: 89271
negative_passages.text: If $F$ is a field, then $F[x]$ is a principal ideal domain proof question
negative_passages.docid: 80961
negative_passages.text: Prove that a commutative ring $F$ is a field iff $F[x]$ is a Principal Ideal Domain
negative_passages.docid: 138130
negative_passages.text: If $I$ is a maximal ideal of $R$, why is $R/I$ a field?
negative_passages.docid: 95408
negative_passages.text: Give an example to show that a factor ring of a ring with divisors of 0 may be an integral domain
negative_passages.docid: 119699
negative_passages.text: Examples of Cohen-Macaulay integral domains
negative_passages.docid: 116943
negative_passages.text: Proof for maximal ideals in $\mathbb{Z}[x]$
negative_passages.docid: 140553
negative_passages.text: An ideal which is not finitely generated