Hereditary ring

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In mathematics, 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 (semi-)hereditary (all (finitely generated) submodules of projective left R-modules are projective) and right (semi-)hereditary are sometimes used. A ring can be left hereditary but not right hereditary, and vice versa.

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