Levitzki radical
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| Levitzki radical | |
|---|---|
| Name | Levitzki radical |
| Field | Algebra |
| Subfield | Ring theory |
| Introduced | 1951 |
| Named after | Alexander Levitzki |
| Related | Jacobson radical, Baer radical, nilradical, prime radical |
Levitzki radical. The Levitzki radical is a radical concept in ring theory that assigns to each associative ring a largest locally nilpotent ideal; it generalizes classical nilpotency notions and interacts with structure theory for rings, modules, algebras, and representations. It plays a central role in the study of Jacobson radical behavior in noncommutative settings, connects to results by Isaac Jacobson, Nathan Jacobson, Amitsur, and appears in classification problems related to Artinian rings, Noetherian rings, and PI-algebras.
Definition and basic properties
The Levitzki radical of a ring R is defined as the sum of all nilpotent ideals of R, equivalently the largest locally nilpotent ideal; it is a two-sided ideal stable under ring homomorphisms and extensions, and it is contained in the Jacobson radical and the Baer radical. For rings satisfying a polynomial identity such as Matrix rings over commutative rings, the Levitzki radical coincides with other radicals studied by Levitzki, Kaplansky, and Kurosh. Basic properties include hereditary behavior for quotients by semiprime ideals, functoriality for ring morphisms between algebras over fields like C and Q, and compatibility with direct limits encountered in constructions related to Grothendieck categories and Von Neumann regular rings.
Historical background and naming
The notion is named after Alexander Levitzki and was developed amid mid-20th-century advances by figures such as Jacob Levitzki, Nathan Jacobson, Israel Herstein, Jacob Levitzki (mathematician) (note: different transliterations), Emil Artin, and Amitsur. Early work on nilpotent ideals traces to investigations by Kurt Hensel-era algebraists and later formalizations appeared in papers influenced by the school of Noether, Hopf, and E. Noether collaborators, while applications and structural theorems were advanced by researchers at institutions such as University of Chicago, Hebrew University of Jerusalem, Institute for Advanced Study, and Princeton University.
Characterizations and equivalent formulations
Several equivalent formulations relate the Levitzki radical to other constructs: as the intersection of annihilators of faithful prime modules appearing in the literature of Goldie and Goldman; as the largest ideal whose quotient yields a semiprime ring in the sense used by Bourbaki and Chevalley; and via descending chain conditions connected with Artinian and Noetherian hypotheses invoked in theorems by Levitzki and Hopkins. In the setting of PI-algebras, Levitzki-type results tie to the Kaplansky theorem and the Posner theorem, while connections to the Brown–McCoy radical and the Semiprime ring notion yield equivalent perspectives used by Martindale and Small.
Examples and computations
Concrete instances include classical computations for upper triangular Matrix rings over fields such as R and C where the Levitzki radical equals the strictly upper triangular matrices, examples drawn from group algebras like Group algebras kG for nilpotent p-groups studied by Burnside-style arguments, and constructions within Incidence algebras over posets related to work by Rota. Commutative examples reduce to the nilradical in contexts involving rings such as Z and polynomial rings k[x], while noncommutative illustrations feature Weyl algebras, Enveloping algebras of Lie algebras, and algebras satisfying polynomial identities studied by Regev and Procesi. Calculations often use annihilator chains and Engel-type criteria inspired by results of Engel and techniques from representation theory of Lie algebras.
Relationships with other radicals and ideals
The Levitzki radical sits between several classical radicals: it is contained in the Jacobson radical and contains the nilpotent radical in many contexts; it compares with the Baer radical, prime radical, and the Brown–McCoy radical according to structural hypotheses such as semiprimeness, primeness, and polynomial identities. Results by Wedderburn, Artin–Wedderburn theorem contexts, and modifications by Martindale characterize when these radicals coincide for rings satisfying chain conditions like ACC or DCC on ideals. Interactions with Idempotent-generated ideals, primitive ideal theory from Harish-Chandra-inspired representation contexts, and localization techniques used by Gabriel further clarify placement within the lattice of ideals.
Applications and significance in ring theory
The Levitzki radical is used to decompose rings into "radical" and "semisimple" parts in analogues of the Wedderburn decomposition, underpin structural classification results for Artinian and Noetherian algebras, and inform the study of modules such as simple modules and injective modules. It appears in proofs of theorems about nilpotence of ideals in PI-algebras, in establishment of Engel conditions for associative algebras mirroring Lie algebra theory, and in decomposition techniques applied in the representation theory of algebraic groups and quantum groups. The Levitzki radical also guides algorithmic approaches in computational algebra systems used by researchers at institutions like MIT and Stanford University for testing nilpotency, and it influences ongoing research in noncommutative geometry and categorical generalizations pursued at centers including IHÉS and Max Planck Institute for Mathematics.