query: If $B \setminus A$ is closed under multiplication then $A$ is integrally closed in $B$.
positive_passages.docid: 157420
positive_passages.text: Question about Subrings and integraly closed
negative_passages.docid: 74312
negative_passages.text: If Y is complete then B(X,Y) is complete
negative_passages.docid: 115606
negative_passages.text: Proofs involving $ (A\setminus B) \cup (A \cap B)$
negative_passages.docid: 83714
negative_passages.text: $A \in B$ vs. $A \subset B$ for proofs
negative_passages.docid: 127418
negative_passages.text: If ring $B$ is integral over $A$, then an element of $A$ which is a unit in $B$ is also a unit in $A$.
negative_passages.docid: 99678
negative_passages.text: Is a sphere a closed set?
negative_passages.docid: 46698
negative_passages.text: Showing a set is closed under inverses
negative_passages.docid: 64656
negative_passages.text: how do you prove that $A\times (B\setminus C) = (A\times B) \setminus (A\times C)$?
negative_passages.docid: 144360
negative_passages.text: For any sets $A$ and $B$, show that $(B\smallsetminus A)\cup A=B \iff A\subseteq B$.
negative_passages.docid: 144266
negative_passages.text: set theory proof of $A\cap B = A \setminus(A\setminus B)$
negative_passages.docid: 10180
negative_passages.text: How to prove that this set is closed?
negative_passages.docid: 138199
negative_passages.text: Proof explanation $A\setminus B = A \setminus B \cap A$
negative_passages.docid: 99341
negative_passages.text: Basic question: closure of irreducible sets
negative_passages.docid: 22854
negative_passages.text: Let $A , B\subseteq\mathbb{R}$. If $A$ is closed and $B$ is compact, is $A\cdot B$ closed?
negative_passages.docid: 85782
negative_passages.text: UFDs are integrally closed
negative_passages.docid: 144383
negative_passages.text: The closure of a connected set in a topological space is connected
negative_passages.docid: 21114
negative_passages.text: $S=\{(n,{1\over n}):n\in\mathbb{N}\}$ is closed in $X$?