query: $R$ be a commutative ring with unity such that every prime ideal contains no non-zero zero divisor , then is $R$ an integral domain?
positive_passages.docid: 22544
positive_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: 3129
negative_passages.text: A commutative ring is a field iff the only ideals are $(0)$ and $(1)$
negative_passages.docid: 14055
negative_passages.text: How to show that a finite commutative ring without zero divisors is a field?
negative_passages.docid: 116453
negative_passages.text: Is any quotient of a Euclidean domain by a prime ideal a Euclidean domain?
negative_passages.docid: 74314
negative_passages.text: Noetherian ring and the same prime divisors
negative_passages.docid: 67342
negative_passages.text: Why does a minimal prime ideal consist of zerodivisors?
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: 80232
negative_passages.text: Suppose that R is a commutative ring with unity such that for each $a$ in $R$ there is an integer $n > 1\mid a^n =a$. Every prime ideal is maximal?
negative_passages.docid: 79957
negative_passages.text: If $R$ is an infinite ring, then $R$ has either infinitely many zero divisors, or no zero divisors
negative_passages.docid: 135699
negative_passages.text: Questions about a commutative ring with exactly three ideals
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: 107454
negative_passages.text: Why D/I is finite if I is a non-zero ideal of the ring of integers of a number field?
negative_passages.docid: 77198
negative_passages.text: Showing that the characteristic of a commutative ring R without zero divisors is 0 or prime
negative_passages.docid: 21604
negative_passages.text: A complete ring with respect to an ideal
negative_passages.docid: 69353
negative_passages.text: Show that a ring with only trivial right ideals is either a division ring or $|R|=p$ and $R^2=\{0\}$. Why would $R$ be finite?
negative_passages.docid: 141659
negative_passages.text: Every field is an integral domain
negative_passages.docid: 135152
negative_passages.text: Non-UFD integral domain such that prime is equivalent to irreducible?