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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity.\nThe ring $R$ is *left semihereditary* if every finitely generated [[left ideal|5N2CJZ47D4]] of $R$ is a [[projective module|BFFA03GE3V]] as a left $R$-module.\n*Right semihereditary* is defined in the same way with right ideals, and $R$ is *semihereditary* if it is both.\n\n## Remarks\n\nExample: every [[hereditary ring|89VBQH8HNS]] is semihereditary, since the condition is asked only of some of the left ideals.\nA non-hereditary example is the ring $\\overline{\\Z}$ of all algebraic integers.\nIt is a [[Bézout domain|HGEX40P0Z0]], so every finitely generated ideal is principal, hence free of rank $0$ or $1$, hence projective.\nIt is not Noetherian, because $2^{1/2}, 2^{1/4}, 2^{1/8}, \\dots$ generate a strictly increasing chain of principal ideals; and a hereditary domain is a Dedekind domain, which is Noetherian.\nNon-example: the ring $\\Z/4\\Z$ is not semihereditary, because its ideal $2\\Z/4\\Z$ is finitely generated and not projective.\nEquivalent form: $R$ is left semihereditary if and only if every finitely generated submodule of a projective left $R$-module is projective.\nA commutative integral domain is semihereditary if and only if it is a [[Prüfer domain|2RDAA4PCK8]].\n",
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