On the importance of being primitive

Bell, Jason P.

doi:10.15446/recolma.v53nsupl.84009

Cited by 5 publications

(2 citation statements)

References 54 publications

(77 reference statements)

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“…This condition is known to be satisfied for numerous examples of Noetherian k-algebras, most notably the enveloping algebra U(h) for a finite dimensional k-Lie algebra h as proved independently by Dixmier and Moeglin in [7] and [15]. An ongoing project is to classify algebras satisfying the equivalence, and in [3], a detailed analysis of what is known to date on this subject is given. In this paper, however, we will not be concerned with the classical picture, but with a p-adic analogue.…”

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confidence: 92%

Affinoid Dixmier modules and the deformed Dixmier-Moeglin equivalence

Jones

2021

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The affinoid envelope U (L) K of a free, finitely generated Z p -Lie algebra L, has proven to be useful within the representation theory of compact p-adic Lie groups. Our aim is to further understand the algebraic structure of U (L) K , and to this end, we will define a Dixmier module over U (L) K , and prove that this object is generally irreducible in case where L is nilpotent. Ultimately, we will prove that all primitive ideals in the affinoid envelope can be described in terms of the annihilators of Dixmier modules, and using this, we aim towards proving that these algebras satisfy a version of the classical Dixmier-Moeglin equivalence.Throughout, fix p a prime, K\Q p a finite extension, O the valuation ring of K, π ∈ O a uniformiser. Classical Motivation -The Dixmier-Moeglin equivalenceThis research is partially inspired by a problem in classical non-commutative algebra. Let k be any field of characteristic 0, and let R be a Noetherian k-algebra. We say that a prime ideal P of R is:• Weakly rational if Z(R/P ) is an algebraic field extension of k.• Rational if Z(Q(R/P )) is an algebraic field extension of k, where Q(R/P ) is the Goldie ring of quotients of R/P , in the sense of [16, Theorem 2.3.6].

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Affinoid Dixmier modules and the deformed Dixmier-Moeglin equivalence

Jones

2021

Preprint

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“…We also study prime and primitive ideals of ultragraph Leavitt path algebras in detail. As mentioned in [8], in general, it is very difficult to find all irreducible representations of a ring but often just knowing the annihilators of the simple modules will allow one to prove non-trivial facts about a ring. We, therefore, extend the theory of Chen simple modules to ultragraph Leavitt path algebras and use it to prove the Exel's Effros-Hahn conjecture for ultragraph Leavitt path algebras.…”

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confidence: 99%

On the ideals of ultragraph Leavitt path algebras

Duyen¹,

Nam²

2021

Preprint

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In this article, we provide an explicit description of a set of generators for any ideal of an ultragraph Leavitt path algebra. We provide several additional consequences of this description, including information about generating sets for graded ideals, the graded uniqueness and Cuntz-Krieger theorems, the semiprimeness, and the semiprimitivity of ultragraph Leavitt path algebras, a complete characterization of the prime and primitive ideals of an ultragraph Leavitt path algebra. We also show that every primitive ideal of an ultragraph Leavitt path algebra is exactly the annihilator of a Chen simple module. Consequently, we prove Exel's Effros-Hahn conjecture on primitive ideals in the ultragraph Leavitt path algebra setting (a conclusion that is also new in the context of Leavitt path algebras of graphs).

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Gaps and approximations in the space of growth functions

Greenfeld

2023

Sel. Math. New Ser.

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On the importance of being primitive

Abstract: We give a brief survey of primitivity in ring theory and in particular look at characterizations of primitive ideals in the prime spectrum for various classes of rings.

Cited by 5 publications

References 54 publications

Affinoid Dixmier modules and the deformed Dixmier-Moeglin equivalence

Affinoid Dixmier modules and the deformed Dixmier-Moeglin equivalence

On the ideals of ultragraph Leavitt path algebras

Gaps and approximations in the space of growth functions

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