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.
- 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 and are trivial for .
- 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 is a direct summand of R, and so von Neumann regular rings are left and right semihereditary.
- For any nonzero element x in a domain R, via the map . Hence in any domain, a principal right ideal is free, hence projective. This reflects the fact that domains are right Rickart rings. It follows that if R is a right Bézout domain, so that finitely generated right ideals are principal, then R has all finitely generated right ideals projective, and hence R is right semihereditary. Finally if R is assumed to be a principal right ideal domain, then all right ideals are projective, and R is right hereditary.
- A commutative hereditary integral domain is called a Dedekind domain. A commutative semi-hereditary integral domain is called a Prüfer domain.
- An important example of a (left) hereditary ring is the path algebra of a quiver. This is a consequence of the existence of the standard resolution (which is of length 1) for modules over a path algebra.
- For a left hereditary ring R, every submodule of a free left R-module is isomorphic to a direct sum of left ideals of R and hence is projective.
- Crawley-Boevey, William, Notes on Quiver Representation
- Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294, Zbl 0911.16001
- Osborne, M. Scott (2000), Basic Homological Algebra, Graduate Texts in Mathematics 196, Springer-Verlag, ISBN 0-387-98934-X, Zbl 0948.18001
- Reiner, I. (2003), Maximal Orders, London Mathematical Society Monographs. New Series 28, Oxford University Press, ISBN 0-19-852673-3, Zbl 1024.16008
- Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge: Cambridge University Press, ISBN 0-521-43500-5, Zbl 0797.18001
|This abstract algebra-related article is a stub. You can help Wikipedia by expanding it.|