query: $R$ is integral domain
positive_passages.docid: 154495
positive_passages.text: Prove that a nontrivial subring of a ring that is a domain has the same identity
negative_passages.docid: 17161
negative_passages.text: Can anyone make me understand in simple language why the second condition for being an Euclidean domain is superfluous?
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: 134458
negative_passages.text: Isomorphisms preserving integral domains and fields
negative_passages.docid: 128550
negative_passages.text: Algebraic field extension and intermediate integral domain
negative_passages.docid: 123024
negative_passages.text: A Noetherian integral domain is a UFD iff $(f):(g)$ is principal
negative_passages.docid: 57031
negative_passages.text: Ring of fractions of $R$ is integral over $R$
negative_passages.docid: 71479
negative_passages.text: Is there some elementary proof of invariance of domain?
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: 67699
negative_passages.text: Integral domain that is integrally closed, one-dimensional and not noetherian
negative_passages.docid: 92420
negative_passages.text: Characteristic of  a Non-unital Integral Ring
negative_passages.docid: 3037
negative_passages.text: Integral domain without unity has prime characteristic?
negative_passages.docid: 13174
negative_passages.text: Show that $A[X]/(aX+b)$ is an integral domain
negative_passages.docid: 49303
negative_passages.text: In a noetherian integral domain every non invertible element is a product of irreducible elements
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: 96132
negative_passages.text: $R$ integral domain : $u\in R^*, a \text{ is prime} \iff au \text{ is prime}$
negative_passages.docid: 119699
negative_passages.text: Examples of Cohen-Macaulay integral domains