Inmathematics, especially in the area ofabstract algebra known asmodule theory, aringR is calledhereditary if allsubmodules ofprojective modules overR are again projective. If this is required only forfinitely generated submodules, it is calledsemihereditary.

For anoncommutative ringR, the termsleft hereditary andleft 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 projectiveleftR-modules must be projective, and similarly to be right (semi-)hereditary all (finitely generated) submodules of projectiverightR-modules must be projective. It is possible for a ring to be left (semi-)hereditary but not right (semi-)hereditary and vice versa.

• The ringR is left (semi-)hereditaryif and only if all (finitely generated)left ideals ofR are projective modules.^[1][2]
• The ringR is left hereditary if and only if all leftmodules haveprojective resolutions of length at most 1. This is equivalent to saying that the leftglobal dimension is at most 1. Hence the usualderived functors such as${\displaystyle {\mathrm{E}\mathrm{x}\mathrm{t}}_R^i}$ and${\displaystyle {\mathrm{T}\mathrm{o}\mathrm{r}}_i^R}$ are trivial for${\displaystyle i>1}$.

• Semisimple rings are left and right hereditary via the equivalent definitions: all left and right ideals are summands ofR, and hence are projective. By a similar token, in avon Neumann regular ring every finitely generated left and right ideal is a direct summand ofR, and so von Neumann regular rings are left and right semihereditary.
• For any nonzero elementx in adomainR,${\displaystyle R\cong xR}$ via the map${\displaystyle r\mapsto xr}$. Hence in any domain, aprincipal right ideal isfree, hence projective. This reflects the fact that domains are rightRickart rings. It follows that ifR is a rightBézout domain, so that finitely generated right ideals are principal, thenR has all finitely generated right ideals projective, and henceR is right semihereditary. Finally ifR is assumed to be aprincipal right ideal domain, then all right ideals are projective, andR is right hereditary.
• A commutative hereditaryintegral domain is called aDedekind domain. A commutative semi-hereditary integral domain is called aPrüfer domain.
• An important example of a (left) hereditary ring is thepath algebra of aquiver. This is a consequence of the existence of the standard resolution (which is of length 1) for modules over a path algebra.
• Thetriangular matrix ring is right hereditary and left semi-hereditary but not left hereditary.
• IfS is a von Neumann regular ring with an idealI that is not a direct summand, then the triangular matrix ring is left semi-hereditary but not right semi-hereditary.

• For a left hereditary ringR, every submodule of afree leftR-module isisomorphic to the direct sum of left ideals ofR and hence is projective.^[2]

• Crawley-Boevey, William,Notes on Quiver Representation(PDF), archived fromthe original(PDF) on 2 May 2003
• 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, vol. 196, Springer-Verlag,ISBN 0-387-98934-X,Zbl 0948.18001
• Reiner, I. (2003),Maximal Orders, London Mathematical Society Monographs. New Series, vol. 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, vol. 38, Cambridge:Cambridge University Press,ISBN 0-521-43500-5,Zbl 0797.18001

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