A note on the Diximier-Mouglin Equivalence in Leavitt path algebras of arbitrary graphs over a field

2021/12/18 by Rangaswamy, Kulumani M.
#16S88 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2112.10016

Abstract

The Dixmier-Moeglin Equivalence (for short, the DM-equivalence) is the equivalence of three distinguishing properties of prime ideals in a non-commutative algebra A. These properties are of (i) being primitive, (ii) being rational, and (iii) being locally closed in the Zariski topology of Spec(A). The DM-equivalence holds in many interesting algebras over a field. Recently, it was shown that the prime ideals of a Leavitt path algebra of a finite graph satisfy the DM-equivalence, In this note, we investigate the occurrence of the DM-equivalence in a Leavitt path algebra L of an arbitrary directed graph E. Our analysis shows that locally closed prime ideals of L satisfy interesting equivalent properties such as being strongly primitive and completely irreducible. Examples illustrate the results and the constraints.

Related

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• A Survey on The Ideal Structure of Leavitt Path Algebras
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