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  "body": "\n## Summary\n\nA quasi-Frobenius ring is a [[ring|0B1261B5ED]] that is Noetherian and [[self-injective|KQKJENEMSN]] on one side; it is then Artinian and self-injective on both sides.\nOver such a ring a module is projective if and only if it is injective.\n[[Frobenius algebras|8EVP1K2CZH]], group algebras of finite groups over fields and the rings $\\Z/n\\Z$ are quasi-Frobenius.\nThe class generalizes Frobenius algebras from algebras over a field to arbitrary rings.\n\n## Explanations\n\n### L1 human\n\nFor most rings, projective modules and injective modules are very different kinds of modules.\nIn a quasi-Frobenius ring they are the same modules.\nThe rings $\\Z/n\\Z$ are examples: they are finite, hence Noetherian, and they are injective over themselves.\nThe ring $\\Z$ is not quasi-Frobenius: $\\Z$ is projective over itself but not injective.\n",
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