Hereditary ring

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Description:
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 <math>mathrm_R^i</math> and <math>mathrm_i^R</math> are trivial for <math>i>1</math>.


Examples

  • Semisimple rings are easily seen to be left and right hereditary via the equivalent definitions: all left and right ideals are summands of R, and hence are projective. By a similar token, in a von Neumann regular ring every finitely generated left and right ideal......
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