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hereditary ring : ウィキペディア英語版
hereditary ring
In mathematics, especially in the area of abstract algebra known as module theory, a ring ''R'' is called hereditary if all submodules of projective modules over ''R'' are again projective. If this is required only for finitely generated submodules, it is called semihereditary.
For a noncommutative ring ''R'', the terms left hereditary and left semihereditary and their right hand versions are used to distinguish the property on a single side of the ring. To be left (semi-)hereditary, all (finitely generated) submodules of projective ''left'' ''R''-modules must be projective, and to be right (semi-)hereditary all (finitely generated) submodules of projective right submodules must be projective. It is possible for a ring to be left (semi-)hereditary but not right (semi-)hereditary, and vice versa.
==Equivalent definitions==

* The ring ''R'' is left (semi-)hereditary if and only if all (finitely generated) left ideals of ''R'' are projective modules.
* The ring ''R'' is left hereditary if and only if all left modules have projective resolutions of length at most 1. Hence the usual derived functors such as \mathrm_R^i and \mathrm_i^R are trivial for i>1.

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