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I see the following notation: for a ring $A,$ a prime ideal $\mathfrak q,$ and $x\in A$. $$ (\mathfrak q :x) = (1). $$ The question is: what does $(\mathfrak q :x)$ mean? The ideal generated by $\mathfrak q ,x?$ Then why do we need to use a colon?

Ma Joad
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  • Notation can change from author to author, but the colon that I am used to between ideals is this one. – plop Jun 16 '21 at 12:38
  • My guess is that the definition is the one given by @plop and that the statement should have been $$(\mathfrak q:x)=\begin{cases}\mathfrak q&\text{if }x\notin\mathfrak q\ (1)&\text{if }x\in\mathfrak q\end{cases}$$ –  Jun 16 '21 at 12:54

1 Answers1

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For a ring $R$, my guess would be

$$(\mathfrak{q} : x) = \{r \in R : rx\in \mathfrak{q}\} $$

Since I've seen the same notation used for subsets $I, J\subset R$

$$(I : J) = \{r\in R: rJ\subseteq I\}$$

SeraPhim
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  • There's a problem: if $\mathfrak q$ is prime, we have $(\mathfrak q : x) = (1)$ iff $x\in\mathfrak q$. So we should know whether the above equality is a hypothesis on $x$. – Bernard Jun 16 '21 at 12:44
  • @Bernard Good point, I did think that just after posting. Perhaps this was better posted as a comment rather than an answer. – SeraPhim Jun 16 '21 at 12:47
  • Not necessarily, as it may help other people who are not accustomed to this notation. – Bernard Jun 16 '21 at 12:54