query: Let R be a commutative ring with identity. Prove: every nilpotent is inside every prime ideal.
positive_passages.docid: 15904
positive_passages.text: Prime ideal and nilpotent elements
negative_passages.docid: 116943
negative_passages.text: Proof for maximal ideals in $\mathbb{Z}[x]$
negative_passages.docid: 133878
negative_passages.text: Ring of order $p^2$ is commutative.
negative_passages.docid: 74559
negative_passages.text: $A$ and $B$ are ideals of a ring ${R}$ such that $A\cap B=\{0\}$. Prove that $st=0$ for every $s\in A, t\in B$.
negative_passages.docid: 131671
negative_passages.text: Let R be a ring such that $a^2 = a$ , $\forall a$ $\in R$ . Prove that R is commutative.
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: 29019
negative_passages.text: Ideals in direct product of rings
negative_passages.docid: 149753
negative_passages.text: The structure of a Noetherian ring in which every element is an idempotent.
negative_passages.docid: 83100
negative_passages.text: Prove that the center of a ring is a subring.
negative_passages.docid: 156878
negative_passages.text: If every commutator is idempotent, then the ring is commutative
negative_passages.docid: 69649
negative_passages.text: If an ideal contains the multiplicative identity, then it is the whole 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: 67669
negative_passages.text: Prime ideals of a finite direct product ring
negative_passages.docid: 75520
negative_passages.text: If $a^3 =a$ for all $a$ in a ring $R$, then $R$ is commutative.
negative_passages.docid: 129302
negative_passages.text: Proving whether ideals are prime in $\mathbb{Z}[\sqrt{-5}]$
negative_passages.docid: 81296
negative_passages.text: If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal
negative_passages.docid: 72230
negative_passages.text: Equivalence of definitions of prime ideal in commutative ring