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  "body": "\n## Summary\n\nA left semihereditary ring is a [[ring|0B1261B5ED]] in which every finitely generated left ideal is a [[projective module|BFFA03GE3V]].\nEvery [[hereditary ring|89VBQH8HNS]] is semihereditary, and the two notions agree for left Noetherian rings.\nCommutative semihereditary domains are exactly the [[Prüfer domains|2RDAA4PCK8]], and every [[von Neumann regular ring|DXX7GNY3HE]] is semihereditary.\n\n## Explanations\n\n### L1 human\n\nSemihereditary rings ask for less than hereditary rings: only the finitely generated ideals need to be projective.\nThis weaker condition matters for rings that are not Noetherian, where some ideals need infinitely many generators.\nThe ring of all algebraic integers is a good example.\nEvery finitely generated ideal in it is principal, hence projective, so it is semihereditary.\nIt is not hereditary, because it has an ideal that is not finitely generated and not projective.\n",
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