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  "body": "\n## Definition\n\nLet $R$ be a ring and let $M$ be a left [[$R$-module|09VDS2W1E8]].\nA [[submodule|0RGZ10DFPG]] $N$ of $M$ is *essential* in $M$ if $N \\cap L \\ne 0$ for every nonzero submodule $L$ of $M$.\nIn that case $M$ is called an *essential extension* of $N$.\n\n## Remarks\n\nEquivalently, for every nonzero $m \\in M$ there is $r \\in R$ with $rm \\in N$ and $rm \\ne 0$.\nNotation: $N \\subseteq_e M$ or $N \\trianglelefteq_e M$, depending on the author.\nExample: for $n \\ne 0$, the submodule $n\\Z$ is essential in the $\\Z$-module $\\Z$.\nThe check: a nonzero submodule $a\\Z$ of $\\Z$ contains $na \\ne 0$, which lies in $n\\Z$.\nNon-example: $\\Z \\oplus 0$ is not essential in $\\Z \\oplus \\Z$, since the nonzero submodule $0 \\oplus \\Z$ meets it only in $0$.\nA nonzero direct summand $N \\ne M$ is never essential, since its complement meets it in $0$.\n",
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  "msc": [
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  "name": "Essential submodule",
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