query: $Rad(I)$ is an ideal of $R$
positive_passages.docid: 20877
positive_passages.text: The set of all nilpotent elements is an ideal
negative_passages.docid: 21604
negative_passages.text: A complete ring with respect to an ideal
negative_passages.docid: 50809
negative_passages.text: Proving $R-S$ contains a prime ideal when $S$ is a multiplicative set
negative_passages.docid: 31585
negative_passages.text: Primary ideals of $\mathbb{Z}$
negative_passages.docid: 133046
negative_passages.text: Showing $[L,rad(L)]$ is nilpotent.
negative_passages.docid: 87151
negative_passages.text: Primary ideals in Noetherian rings
negative_passages.docid: 47134
negative_passages.text: Ideals generated by two elements
negative_passages.docid: 144250
negative_passages.text: subideals of an ideal
negative_passages.docid: 138168
negative_passages.text: Prove: The pre-image of an ideal is an ideal.
negative_passages.docid: 34183
negative_passages.text: An equivalent condition with $\{0\}$ being the only nilpotent ideal
negative_passages.docid: 29126
negative_passages.text: Maximal ideal in the ring of continuous functions from $\mathbb{R} \to \mathbb{R}$
negative_passages.docid: 29019
negative_passages.text: Ideals in direct product of rings
negative_passages.docid: 110984
negative_passages.text: Ideals and prime ideals in a commutative ring.
negative_passages.docid: 464
negative_passages.text: Prove that the ideal $(X_1-a_1,...,X_n-a_n)$ is maximal in $K[X_1,\dots,X_n]$
negative_passages.docid: 11109
negative_passages.text: The ideal generated by a non-compact operator
negative_passages.docid: 16945
negative_passages.text: Finding the Ideals of a Direct Product of Rings
negative_passages.docid: 58640
negative_passages.text: $R$ is an algebra over an infinite field. If $\exists$ ideals s.t. $J\subseteq \bigcup_{k=1}^nI_k$ then $J\subseteq I_k$ for some $k$