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Let $R$ be a local principal ideal domain (PID) with only two prime ideals $0$ and $P$, and let $M$ be an $R$-module. Let for $r\in R$ and $m\in M$, $rm\not=0$. Now if $E(rm)$ is a fixed injective envelope for $rm$, then can we find an injective envelope for $m$ containing $E(rm)$?

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    $\begingroup$ Are you sure you are looking for an "invective envelop" ? Maybe, it is also worth saying what is an injective envelop for an element of a module. $\endgroup$
    – A.B.
    Sep 17, 2021 at 18:09
  • $\begingroup$ I am looking for an injective envelope $A$ for $m$ such that $E(rm)\subseteq A$. $\endgroup$
    – Antony
    Sep 18, 2021 at 15:15
  • $\begingroup$ What is “an injective envelope for an element”? The standard definition is for an injective envelope of a module. $\endgroup$
    – rschwieb
    Oct 3, 2021 at 15:39

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