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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity.\nThe ring $R$ is *left self-injective* if $R$, viewed as a left [[module|0NZ8JA25WK]] over itself by left multiplication, is an [[injective module|85H7PRHDPZ]].\nIt is *right self-injective* if $R$ is injective as a right module over itself, and *self-injective* if it is both.\n\n## Remarks\n\nExample: $R = \\Z/n\\Z$ for $n \\ge 1$ is self-injective.\nBy Baer's criterion it suffices to extend maps $f$ from an ideal $d\\Z/n\\Z$, with $d \\mid n$, to $R$.\nThe image $a = f(d)$ satisfies $(n/d) a = 0$, so $a = db$ for some $b \\in R$, and $x \\mapsto bx$ extends $f$.\nNon-example: $\\Z$ is not self-injective.\nThe map $2\\Z \\to \\Z$, $2m \\mapsto m$, has no extension $g \\colon \\Z \\to \\Z$, since $2 g(1) = 1$ has no solution in $\\Z$.\nLeft and right self-injectivity are different conditions in general.\nA ring is [[quasi-Frobenius|KP1PP7A6JR]] if and only if it is left Noetherian and left self-injective.\n",
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