Products and Intersections of Prime-Power Ideals in Leavitt Path Algebras

TL;DR

This paper investigates the structure of ideals in Leavitt path algebras, proving uniqueness of factorizations into prime-power ideals and characterizing completely irreducible ideals.

Contribution

It establishes the uniqueness of ideal factorizations into prime-power ideals and characterizes completely irreducible ideals in Leavitt path algebras.

Findings

Unique factorization of ideals into prime-power ideals.

Characterization of completely irreducible ideals as prime-power ideals.

Identification of ideals that factor into completely irreducible ideals.

Abstract

We continue a very fruitful line of inquiry into the multiplicative ideal theory of an arbitrary Leavitt path algebra L. Specifically, we show that factorizations of an ideal in L into irredundant products or intersections of finitely many prime-power ideals are unique, provided that the ideals involved are powers of distinct prime ideals. We also characterize the completely irreducible ideals in L, which turn out to be prime-power ideals of a special type, as well as ideals that can be factored into products or intersections of finitely many completely irreducible ideals.