query: Prove that $x = uy$ for some unit $u \in R^{\times}$ iff $(x)=(y)$ as ideals.
positive_passages.docid: 16023
positive_passages.text: Are associates unit multiples in a commutative ring with $1$?
negative_passages.docid: 36150
negative_passages.text: Ideal in a ring of continuous functions
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: 80209
negative_passages.text: If $X,Y$ are equivalence relations, so is $X \times Y$
negative_passages.docid: 93658
negative_passages.text: Proving that an ideal in a PID is maximal if and only if it is generated by an irreducible
negative_passages.docid: 587
negative_passages.text: Ideal Generated by the Union of Two Ideals
negative_passages.docid: 61103
negative_passages.text: Prove that $\mathbb{|Q| = |Q\times Q|}$
negative_passages.docid: 111129
negative_passages.text: Show that $[K:F]_s = [K:L]_s [L:F]_s$ and $[K:F]_i = [K:L]_i [L:F]_i$.
negative_passages.docid: 17075
negative_passages.text: Show an Ideal is the principal ideal for some polynomial.
negative_passages.docid: 76432
negative_passages.text: $i^{-1} F$ a sheaf if and only if $\varinjlim_{ U \subseteq X \text{ open}, ~ x,y \in U } F(U) \to F_x \times F_y$ is an isomorphism.
negative_passages.docid: 80916
negative_passages.text: How to prove that if $A\subseteq A\times A$, then $A=\varnothing$?
negative_passages.docid: 46746
negative_passages.text: How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$?
negative_passages.docid: 87946
negative_passages.text: Proving that the unities of a ring form a group under multiplication
negative_passages.docid: 72640
negative_passages.text: $(m,n) = (\gcd(m,n))$ as ideals in $\Bbb Z$
negative_passages.docid: 142530
negative_passages.text: Let K be a field, and $I=(XY,(X-Y)Z)⊆K[X,Y,Z]$. Prove that $√I=(XY,XZ,YZ)$.
negative_passages.docid: 155745
negative_passages.text: Ideals of $\mathbb{Z}[X]$
negative_passages.docid: 91180
negative_passages.text: Show $R \setminus S$ is a union of prime ideals