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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity.\nThe ring $R$ is *left hereditary* if every [[left ideal|5N2CJZ47D4]] of $R$ is a [[projective module|BFFA03GE3V]] as a left $R$-module.\nIt is *right hereditary* if every right ideal is projective as a right $R$-module.\nIt is *hereditary* if it is both left and right hereditary.\n\n## Remarks\n\nExample: the ring $\\Z$ is hereditary.\nEach ideal is $n\\Z$ for some $n \\ge 0$; it is $0$ or isomorphic to $\\Z$ through $x \\mapsto x/n$, so it is free and hence projective.\nNon-example: the ring $\\Z/4\\Z$ is not hereditary.\nIt is a local ring, so its projective modules are free.\nThe ideal $2\\Z/4\\Z$ has $2$ elements, while a nonzero free $\\Z/4\\Z$-module has at least $4$, so this ideal is not projective.\nEquivalent forms: $R$ is left hereditary if and only if every submodule of a projective left $R$-module is projective.\nThis holds if and only if the left [[global dimension|GWA0T1WKCN]] of $R$ is at most $1$.\nFor a ring that is Noetherian on both sides, left and right hereditary are equivalent.\nAn integral domain is hereditary if and only if it is a [[Dedekind domain|7NPY2CN5KB]].\nThe path algebra of a finite quiver without oriented cycles over a field is hereditary.\n",
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