query: Let $I$ be an ideal of ring $R$ . If $K$ is ideal
positive_passages.docid: 143073
positive_passages.text: Ideal correspondence
negative_passages.docid: 64484
negative_passages.text: Left ideal $I$ is a direct summand of $R$ iff $I = Rr$, $r^2 = r$
negative_passages.docid: 56278
negative_passages.text: $I$ and $J$ are coprime ideals iff $x \to (x + I, x + J)$ is surjective.
negative_passages.docid: 70320
negative_passages.text: Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is $I=(0)$?
negative_passages.docid: 30714
negative_passages.text: A question on ring homomorphisms and maximal ideals.
negative_passages.docid: 85206
negative_passages.text: Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.
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 $k$
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: 52966
negative_passages.text: Rings, ideals and units-proof
negative_passages.docid: 16602
negative_passages.text: Proving $M_n(I)\lhd M_n(R)$ and proofs about ideal rings.
negative_passages.docid: 100961
negative_passages.text: Ideals in valuation rings
negative_passages.docid: 129302
negative_passages.text: Proving whether ideals are prime in $\mathbb{Z}[\sqrt{-5}]$
negative_passages.docid: 36150
negative_passages.text: Ideal in a ring of continuous functions
negative_passages.docid: 138752
negative_passages.text: Prove the Radical of an Ideal is an Ideal
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: 30327
negative_passages.text: $\ker \phi = (x_1-a_1, ..., x_n-a_n)$ for a ring homomorphism $\phi: R[x_1, ..., x_n] \to R$
negative_passages.docid: 157605
negative_passages.text: $IS \mid JS$ implies that $I\mid J$ where $I$ and $J$ are ideals of a number ring contained in another number ring $S$