query: Show that $S^{-1}A = B$ for integral domains and $S = \{x\in A\setminus\{0\}: x^{-1}\in B\}$
positive_passages.docid: 150170
positive_passages.text: Subrings of fraction fields
negative_passages.docid: 13575
negative_passages.text: Need help for proving that: $f(f^{-1}(A)) ⊆ A$
negative_passages.docid: 26741
negative_passages.text: How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
negative_passages.docid: 68705
negative_passages.text: Prove that $A\ge0, B\ge0$ and $A\ge B$ implies $B^{-1}\ge A^{-1}$
negative_passages.docid: 71479
negative_passages.text: Is there some elementary proof of invariance of domain?
negative_passages.docid: 81775
negative_passages.text: Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$
negative_passages.docid: 24391
negative_passages.text: $\arcsin$ written as $\sin^{-1}(x)$
negative_passages.docid: 11259
negative_passages.text: Prove $R\times S$ is not an integral domain
negative_passages.docid: 123275
negative_passages.text: Prove $[A,B^n] = nB^{n-1}[A,B]$
negative_passages.docid: 59872
negative_passages.text: How to prove $\inf(S)=-\sup(-S)$?
negative_passages.docid: 82097
negative_passages.text: If $A \succeq B$, is it true that $B^{-1} \succeq A^{-1}$?
negative_passages.docid: 33524
negative_passages.text: Prove the following $A \setminus (B\setminus C) = (A\setminus B)\cap (A \cup C)$
negative_passages.docid: 64172
negative_passages.text: Show that if $f$ is analytic in a domain $D$ and $|f(z)|$ is constant in $D$, then the function $f(z)$ is constant in $D$
negative_passages.docid: 1580
negative_passages.text: An integral domain $A$ is exactly the intersection of the localisations of $A$ at each maximal ideal
negative_passages.docid: 33030
negative_passages.text: Proof that $\inf A = -\sup(-A)$
negative_passages.docid: 96132
negative_passages.text: $R$ integral domain : $u\in R^*, a \text{ is prime} \iff au \text{ is prime}$
negative_passages.docid: 139011
negative_passages.text: How to prove $B-A \succeq 0 \Leftrightarrow$ ellipsoid $x^TBx \leq 1$ contains $x^TAx \leq 1$?