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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3$\begingroup$ 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
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$\begingroup$ I am looking for an injective envelope $A$ for $m$ such that $E(rm)\subseteq A$. $\endgroup$– AntonySep 18, 2021 at 15:15
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$\begingroup$ What is “an injective envelope for an element”? The standard definition is for an injective envelope of a module. $\endgroup$– rschwiebOct 3, 2021 at 15:39
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