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  "body": "\n## Summary\n\nAn essential submodule of a [[module|0NZ8JA25WK]] $M$ is a [[submodule|05WDSN597C]] that meets every nonzero submodule of $M$ in a nonzero element.\nOne then says that $M$ is an essential extension of it.\nEssential submodules are \"large\" in a lattice sense, even when they are small in size.\nEssential extensions are the basis of the theory of injective hulls, uniform dimension and Goldie's theorem.\n\n## Explanations\n\n### L1 human\n\nA submodule is essential when it cannot be avoided: every nonzero piece of the big module overlaps it.\nThe even integers $2\\Z$ are essential in $\\Z$, since any nonzero subgroup $a\\Z$ contains the nonzero even number $2a$.\nIn $\\Z \\oplus \\Z$, the first axis $\\Z \\oplus 0$ is not essential, since the second axis meets it only in zero.\nThe rational numbers $\\Q$ are an essential extension of $\\Z$, and $\\Q$ is the injective hull of $\\Z$.\n",
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