query: $R$ integral domain, $P$ projective and injective module $\implies P=0$ or $R$ is field of fractions
positive_passages.docid: 144122
positive_passages.text: A problem about an $R$-module that is both injective and projective.
negative_passages.docid: 11259
negative_passages.text: Prove $R\times S$ is not an integral domain
negative_passages.docid: 109647
negative_passages.text: Ext$_R^n(Q,A)=0=$Tor$_n^R(Q,A)$ where $Q$ is the field of fractions of a domain $R$
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: 13983
negative_passages.text: Show that If $R[X]$ is Euclidean domain then $R$ is a field
negative_passages.docid: 141659
negative_passages.text: Every field is an integral domain
negative_passages.docid: 148111
negative_passages.text: Every $R$-module is projective if and only if every $R$-module is injective
negative_passages.docid: 73344
negative_passages.text: Definition of irreducible polynomial via the units of an integral domain.
negative_passages.docid: 124672
negative_passages.text: Let $f:R\rightarrow S$ be a ring homomorphism. Prove or disprove: if $f$ is onto and $R$ is an integral domain, then $S$ is an integral domain
negative_passages.docid: 98345
negative_passages.text: Characteristic of an integral domain must be either $0$ or a prime number.
negative_passages.docid: 24224
negative_passages.text: How to prove the ring of Laurent polynomials over a field is a principal ideal domain?
negative_passages.docid: 13174
negative_passages.text: Show that $A[X]/(aX+b)$ is an integral domain
negative_passages.docid: 78950
negative_passages.text: Does this map extend to a unique automorphism of the polynomial ring of an integral domain?
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: 112099
negative_passages.text: Polynomial ring with integral coefficients is integral
negative_passages.docid: 16589
negative_passages.text: Field of fractions of integral extension is an algebraic extension
negative_passages.docid: 42170
negative_passages.text: $R$ integral domain where every prime ideal is principal then set of non-principal ideals has maximal element