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  "body": "\n## Summary\n\nA left self-injective ring is a [[ring|0B1261B5ED]] that is an [[injective module|85H7PRHDPZ]] when viewed as a left module over itself.\nRings $\\Z/n\\Z$, [[Frobenius algebras|8EVP1K2CZH]] and products of fields are self-injective, while $\\Z$ is not.\nA left Noetherian left self-injective ring is exactly a [[quasi-Frobenius ring|KP1PP7A6JR]].\n\n## Explanations\n\n### L1 human\n\nEvery ring acts on itself by multiplication, so it is a module over itself.\nInjective modules are those into which maps can always be extended from a submodule to a bigger module.\nA self-injective ring is one for which the ring itself has this extension property.\nFor $\\Z/6\\Z$ this holds: any map from an ideal of $\\Z/6\\Z$ into $\\Z/6\\Z$ is multiplication by some element.\nFor $\\Z$ it fails: the map $2\\Z \\to \\Z$ sending $2$ to $1$ cannot be extended to $\\Z$, since $1$ would have to go to $1/2$.\n",
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