On Pr\"ufer-Like Properties of Leavitt Path Algebras

TL;DR

This paper explores how Leavitt path algebras exhibit properties similar to Pr"ufer and Dedekind domains through their ideal lattices, revealing new algebraic characterizations.

Contribution

It demonstrates that Leavitt path algebras satisfy multiple ideal-theoretic properties characteristic of Pr"ufer and almost Dedekind domains, extending their known structural features.

Findings

Leavitt path algebras satisfy distributive law for ideals.

They are multiplication rings, like Dedekind domains.

They fulfill additional properties related to gcd and lcm of ideals.

Abstract

Pr\"{u}fer domains and subclasses of integral domains such as Dedekind domains admit characterizations by means of the properties of their ideal lattices. Interestingly, a Leavitt path algebra $L$, in spite of being non-commutative and possessing plenty of zero divisors, seems to have its ideal lattices possess the characterizing properties of these special domains. In [8] it was shown that the ideals of $L$ satisfy the distributive law, a property of Pr\"{u}fer domains and that $L$ is a multiplication ring, a property of Dedekind domains. In this paper, we first show that $L$ satisfies two more characterizing properties of Pr\"{u}fer domains which are the ideal versions of two theorems in Elementary Number Theory, namely, for positive integers $a,b,c$, $\gcd(a,b)\cdot\operatorname{lcm}(a,b)=a\cdot b$ and $a\cdot \operatorname{gcd}(b,c)=\operatorname{gcd}(ab,ac)$. We also show that $L$ satisfies a characterizing property of almost Dedekind domains in terms of the ideals whose radicals are prime ideals. Finally, we give necessary and sufficient conditions under which $L$ satisfies another important characterizing property of almost Dedekind domains, namely the cancellative property of its non-zero ideals.