hereditary ring
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| hereditary ring | |
|---|---|
| Name | Hereditary ring |
| Type | Ring-theoretic property |
| Area | Algebra |
| Notable | Emmy Noether, Claude Chevalley, Israel Gelfand |
| Examples | Dedekind domain, matrix ring |
| Related | Artinian ring, Noetherian ring, Auslander–Reiten theory |
hereditary ring A hereditary ring is a ring whose one-sided ideals exhibit particularly well-behaved projective structure: every left ideal (or every right ideal) is projective as a module. This property arises in the classification of rings studied by Emmy Noether-inspired algebraists and appears in connections to Dedekind domain, Prufer domain, and Artinian ring theory. Hereditary rings serve as key examples in module theory, representation theory, and algebraic geometry influenced by work of Claude Chevalley and Israel Gelfand.
Definition and basic properties
A (left) hereditary ring R is a ring for which every left ideal is projective as a left R-module. Equivalently, submodules of projective left R-modules are projective; the mirror notion yields right hereditary rings defined by right ideals. Fundamental properties include: hereditary rings are stable under Morita equivalence arising from Morita equivalence results; they often satisfy chain conditions related to Noetherian ring or Artinian ring hypotheses. If R is hereditary and semiprime, structural decompositions connect to classical theorems attributable to Richard Brauer and Nathan Jacobson.
Examples and classes
Classical examples include principal ideal domains such as Euclid’s prototypical integer rings and all Dedekind domains when considered in the commutative setting: every nonzero ideal is projective. Prufer domains provide non-Noetherian commutative examples studied in the tradition of Emmy Noether and Irving Kaplansky. In noncommutative algebra, full matrix rings over division rings produce hereditary behavior after Morita transport from a division algebra like those arising in Wedderburn–Artin theory. Hereditary artinian rings are precisely the finite direct products of matrix rings over division rings, linking to Artin–Wedderburn theorem. Path algebras of acyclic quivers studied in Gabriel’s work give hereditary examples central to Auslander–Reiten theory. Group algebras over fields for finite groups seldom are hereditary except in restricted cases tied to Maschke's theorem and semisimplicity studied by Issai Schur.
Characterizations and equivalent conditions
Several equivalent formulations are standard: R is left hereditary iff every submodule of a projective left R-module is projective; equivalently, global homological dimension on the left satisfies left global dimension ≤ 1. Other characterizations use homological algebra: Ext^2_R(–,–) vanishes for left modules in line with results by Henri Cartan and Samuel Eilenberg. For left Noetherian rings, hereditary coincides with every torsion-free module being flat, connecting to criteria developed in the literature of I. S. Cohen and Hyman Bass. In the finite-dimensional algebra context, hereditary algebras are characterized by having acyclic Auslander–Reiten quivers as in the classification by Bernard Gabriel.
Homological aspects and invariants
Homological invariants for hereditary rings are particularly tractable: projective dimension of every left R-module is at most one, yielding left global dimension ≤ 1 and vanishing of higher Ext groups beyond degree one. This simplifies computation of K-theory invariants akin to those studied by Daniel Quillen and relations with Algebraic K-theory. Derived categories of hereditary algebras admit explicit descriptions, enabling tilting theory and cluster categories developed by researchers influenced by Bernhard Keller and Idun Reiten. Hochschild cohomology calculations reduce complexity and relate to deformation theory as investigated by Gerhard Hochschild.
Relationships with other ring-theoretic concepts
Hereditary rings intersect numerous classical notions: commutative hereditary domains are Dedekind or Prufer domains, linking to ideal class group phenomena central in work of Richard Dedekind and Leopold Kronecker. In the Artinian setting, hereditary equals semisimple, tying to the Artin–Wedderburn theorem and representation theory of Brauer and Jacobson. Hereditary property implies hereditary torsion theories and interacts with localization techniques used by Emmy Noether and Oscar Zariski. Connections to global dimension place hereditary rings alongside regular rings in the sense used by Alexander Grothendieck in algebraic geometry; in particular, regular local rings have finite global dimension while hereditary rings exhibit the extremal case of dimension one.
Applications and significance in algebra
Hereditary rings play a central role in representation theory of algebras, where path algebras of quivers enable classification of indecomposable modules via Gabriel’s theorem and Auslander–Reiten sequences. They underpin construction of tilting modules and derived equivalences exploited in modern work influenced by Happel, Brenner, and Butler. In commutative algebra and number theory, Dedekind and Prufer examples govern ideal class groups and factorization properties fundamental to algebraic number theory as developed by Richard Dedekind and Ernst Kummer. Computational applications include simplifications in homological algebra algorithms and explicit module classification problems encountered in software influenced by projects such as those by John H. Conway’s computational algebra advocates. Overall, hereditary rings serve as a tractable yet rich class of rings linking classical algebraic structures and contemporary categorical methods.