Hereditary ring

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Hereditary ring: In abstract algebra, a ring R is hereditary if every submodule of a projective R-module is also projective. If this only needs to hold for finitely generated submodules, the ring is called semihereditary. For noncommutative rings, we talk about left and right versions: left hereditary means every submodule of a projective left R-module is projective, and left semihereditary means the same for finitely generated submodules. The right-hand versions apply to right R-modules. It is possible for a ring to be left hereditary but not right hereditary, and vice versa.