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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. This leads to new characterizations of primitive ideals in a Leavitt path algebra of a finite graph. Examples illustrate the results and the constraints.

A note on the Dixmier-Moeglin equivalence in Leavitt path algebras of arbitrary graphs over a field