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  "body": "\n## Summary\n\nA left hereditary ring is a [[ring|0B1261B5ED]] in which every left ideal is a [[projective module|BFFA03GE3V]].\nEquivalently, every submodule of a projective left module is projective, so the left [[global dimension|GWA0T1WKCN]] is at most $1$.\nPrincipal ideal domains, Dedekind domains and path algebras over a field of finite quivers without oriented cycles are hereditary.\nThese rings are the next simplest case after semisimple rings, where every module is projective.\n\n## Explanations\n\n### L1 human\n\nProjective modules are the well-behaved modules, the ones that look like pieces of free modules.\nIn a hereditary ring this good behaviour is inherited by submodules: a submodule of a projective module is again projective.\nThe ring of integers $\\Z$ is the standard example.\nEvery subgroup of a free abelian group is free, so subgroups of projective $\\Z$-modules are projective.\nThe ring $\\Z/4\\Z$ is not hereditary, because its ideal $2\\Z/4\\Z$ is not projective.\n",
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