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	~~The writer~~'s ~~name~~ is ~~Christy Brookins~~. ~~What I adore doing~~ is ~~football~~ but ~~I don~~'t have the ~~time recently~~. ~~Invoicing is what I do~~. ~~Mississippi is where his house~~ is.<br><br>~~Check out my web~~-~~site clairvoyant psychic~~ ([http://~~findyourflirt~~.~~net~~/~~index~~.~~php?m~~=~~member_profile&p~~=~~profile&id~~=~~117823 findyourflirt~~.~~net~~])		In [[mathematics]], especially in the area of [[abstract algebra]] known as [[module theory]], a [[ring (mathematics)\|ring]] ''R'' is called '''hereditary''' if all [[submodule]]s of [[projective module]]s over ''R'' are again projective. If this is required only for [[finitely generated module\|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 module\|finitely generated]]) [[left ideal]]s of ''R'' are projective modules.<ref>{{harvnb\|Lam\|1999\|p=42}}</ref><ref name=R2729>{{harvnb\|Reiner\|2003\|pp=27–29}}</ref>
			* The ring ''R'' is left hereditary if and only if all left modules have [[projective resolution]]s of length at most 1. Hence the usual [[derived functor]]s such as <math>\mathrm{Ext}_R^i</math> and <math>\mathrm{Tor}_i^R</math> are trivial for <math>i>1</math>.

			==Examples==
			* [[Semisimple ring]]s 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 (ring theory)\|domain]] ''R'', <math>R\cong xR</math> via the map <math>r\mapsto xr</math>. Hence in any domain, a principal right ideal is free, hence projective. This reflects the fact that domains are right [[Rickart ring]]s. 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 ideal ring\|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 (mathematics)\|quiver]]. This is a consequence of the existence of the standard resolution (which is of length 1) for modules over a path algebra.

			==Properties==
			* 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.<ref name=R2729/>

			==References==
			{{reflist}}
			*{{citation \| last=Crawley-Boevey \| first=William \| url=http://www.amsta.leeds.ac.uk/~pmtwc/quivlecs.pdf \| title=Notes on Quiver Representation }}
			*{{Citation \| last1=Lam \| first1=Tsit-Yuen \| title=Lectures on modules and rings \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Graduate Texts in Mathematics No. 189 \| isbn=978-0-387-98428-5 \| mr=1653294 \| zbl=0911.16001 \| year=1999}}
			* {{citation \| last = Osborne \| first = M. Scott \| title = Basic Homological Algebra \| series = Graduate Texts in Mathematics \| volume = 196 \| publisher = Springer-Verlag \| date = 2000 \| isbn = 0-387-98934-X \| zbl=0948.18001 }}
			* {{citation \| last=Reiner \| first=I. \| authorlink=Irving Reiner \| title=Maximal Orders \| series=London Mathematical Society Monographs. New Series \| volume=28 \| publisher=[[Oxford University Press]] \| year=2003 \| isbn=0-19-852673-3 \| zbl=1024.16008 }}
			* {{citation \| last=Weibel \| first=Charles A. \| title=An introduction to homological algebra \| series=Cambridge Studies in Advanced Mathematics \| volume=38 \| publisher=[[Cambridge University Press]] \| location=Cambridge \| year=1994 \| isbn=0-521-43500-5 \| zbl=0797.18001 }}

			[[Category:Ring theory]]


			{{Abstract-algebra-stub}}

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2012-08-23T16:00:50Z

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The writer's name is Christy Brookins. What I adore doing is football but I don't have the time recently. Invoicing is what I do. Mississippi is where his house is.<br><br>Check out my web-site clairvoyant psychic ([http://findyourflirt.net/index.php?m=member_profile&p=profile&id=117823 findyourflirt.net])

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