TITLE: Trouble with notation $I:J^{\infty}$
QUESTION [2 upvotes]: I am not sure I understand this notation correctly. The definition says, for a ring $R$ with $I,J$ ideals of $R$, we define $I:J^{\infty}=\cup_{i=1}^{\infty} I:J^i$. Now, $I:J$ is the set of elements of $R$ that multiply $J$ into $I$. So, $I:J^{i+1} \subseteq I:J^i$ and hence $I:J^{\infty}=I:J$. Of course, this is wrong, else we would not need a new definition, but I don't see my error. I would appreciate any help in finding my mistake.
Thanks

REPLY [6 votes]: HINT $\: $ reciprocating reverses containments $\rm\ \ J \supset K\ \Rightarrow\ I:J\: \subset\: I:K\ \ $via $\rm\ \ J\ r \subset I\ \Rightarrow\ K\ r\subset I$
{"set_name": "stack_exchange", "score": 2, "question_id": 49437}