query: Proof that every Principal ideal domain is Noetherian
positive_passages.docid: 92360
positive_passages.text: Every principal ideal domain satisfies ACCP.
negative_passages.docid: 145351
negative_passages.text: Ring of formal power series over a principal ideal domain is a unique factorisation domain
negative_passages.docid: 108896
negative_passages.text: Prime ideal in a polynomial ring over an integrally closed domain
negative_passages.docid: 74314
negative_passages.text: Noetherian ring and the same prime divisors
negative_passages.docid: 30029
negative_passages.text: Can we always choose the generators of an ideal of a Noetherian ring to be homogeneous?
negative_passages.docid: 26805
negative_passages.text: Non-Noetherian 0-dimensional ring
negative_passages.docid: 94532
negative_passages.text: How to prove a commutative, with unit, Noetherian ring $A$ only has finitely many minimal prime ideals via the following step?
negative_passages.docid: 46717
negative_passages.text: If $M$ is a Noetherian $R$-module, then $R/\text{Ann}(M)$ is a Noetherian ring
negative_passages.docid: 154495
negative_passages.text: Prove that a nontrivial subring of a ring that is a domain has the same identity
negative_passages.docid: 85206
negative_passages.text: Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.
negative_passages.docid: 97902
negative_passages.text: In a integral domain every prime element is irreducible
negative_passages.docid: 151018
negative_passages.text: Proving that any finite integral domain is a field.
negative_passages.docid: 119056
negative_passages.text: If in a UFD every maximal ideal is principal then it is a PID
negative_passages.docid: 62227
negative_passages.text: Is a product of two Noetherian schemes over Spec $\mathbb Z$ a Noetherian scheme?
negative_passages.docid: 94819
negative_passages.text: Prove that $\frac{\mathbb{C}[x,y]}{\langle x^2-y^2-1\rangle}$ is an integral domain such that all the nonzero prime ideals are maximal.
negative_passages.docid: 113152
negative_passages.text: Prove Principal Ideal Domain from Bezout's condition, and terminating divisibility chain
negative_passages.docid: 43103
negative_passages.text: Unique factorization domain and principal ideals