Hereditary ring
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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 (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.
In particular, one consequence (in fact, an equivalent condition) of the (left) hereditary property is that all (left) modules have projective resolutions of length at most 1. Hence the usual derived functors such as 
and 
are trivial for i > 1.
[edit] Examples
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.
Principal ideal domains give another class of (commutative) hereditary rings. This follows because a submodule of a free module over a principal ideal domain is again free. Hence every module has a free resolution of length 1, and a fortiori, a projective resolution of length 1. A commutative hereditary integral domain is called a Dedekind domain. A commutative semi-hereditary integral domain is called a Prüfer domain.
[edit] References
- Crawley-Boevey, William, Notes on Quiver Representations
- Osborne, M. Scott (2000), Basic Homological Algebra, Graduate Texts in Mathematics, 196, Springer-Verlag, ISBN 0-387-98934-X
- Weibel, Charles A., An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994. xiv+450 pp. ISBN 0-521-43500-5; 0-521-55987-1
