query: All prime ideals are invertible $\Rightarrow$ Dedekind domain
positive_passages.docid: 150239
positive_passages.text: In an integral domain every non-zero prime ideal is invertible implies every non-zero proper ideal is invertible.
negative_passages.docid: 72230
negative_passages.text: Equivalence of definitions of prime ideal in commutative ring
negative_passages.docid: 13901
negative_passages.text: Finitely many prime ideals lying over $\mathfrak{p}$
negative_passages.docid: 117536
negative_passages.text: Why is the localization at a prime ideal a local ring?
negative_passages.docid: 21946
negative_passages.text: What do prime ideals in $k[x,y]$ look like?
negative_passages.docid: 34807
negative_passages.text: Localization at finitely many minimal prime ideals
negative_passages.docid: 121986
negative_passages.text: Do the Laurent polynomials over $\mathbb{Z}$ form a principal ideal domain?
negative_passages.docid: 75899
negative_passages.text: Principal ideal domain with finitely many ideals
negative_passages.docid: 74558
negative_passages.text: Localization of ideals at all primes
negative_passages.docid: 113152
negative_passages.text: Prove Principal Ideal Domain from Bezout's condition, and terminating divisibility chain
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: 22544
negative_passages.text: Let R be a commutative ring, and let P be a prime ideal of R. Suppose that P has no nontrivial zero divisors in it. Show that R is an integral domain.
negative_passages.docid: 74370
negative_passages.text: $P$ is a prime ideal of $R$ iff $R/P$ is an integral domain. : $P≠R$
negative_passages.docid: 145594
negative_passages.text: Splitting of prime ideals in algebraic extensions
negative_passages.docid: 109199
negative_passages.text: Prove that every nonzero prime ideal is maximal in $\mathbb{Z}[\sqrt{d}]$
negative_passages.docid: 40678
negative_passages.text: Image of an ideal under a surjective ring homomorphism is an ideal
negative_passages.docid: 53367
negative_passages.text: In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal. What could be wrong in this approach?