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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity.\nThe ring $R$ is *quasi-Frobenius* if:\n\n1. $R$ is left Noetherian, that is, $R$ is a [[Noetherian module|AK1ZRCXR32]] as a left module over itself;\n2. $R$ is [[left self-injective|KQKJENEMSN]].\n\n## Remarks\n\nThe definition is left-right symmetric: a ring satisfying the two clauses is also right Noetherian and right self-injective, and it is left and right Artinian.\nExample: $\\Z/n\\Z$ for $n \\ge 1$.\nIt is finite, so clause 1 holds, and it is self-injective by Baer's criterion, so clause 2 holds.\nNon-example: $\\Z$ is Noetherian but not self-injective, since the map $2\\Z \\to \\Z$, $2m \\mapsto m$, does not extend to $\\Z$; clause 2 fails.\nNon-example: $k[x, y]/(x^2, xy, y^2)$ over a field $k$ is Artinian, hence Noetherian, but not self-injective, so clause 2 fails.\nEquivalent forms: $R$ is quasi-Frobenius if and only if every projective left $R$-module is injective, if and only if every injective left $R$-module is projective.\nEvery [[Frobenius algebra|8EVP1K2CZH]] is quasi-Frobenius.\n",
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