On Prufer-like properties of Leavitt path algebras

Abstract

Prufer 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 noncommutative and possessing plenty of zero divisors, seems to have its ideal lattices possess the characterizing properties of these special domains. In [The multiplicative ideal theory of Leavitt path algebras, J. Algebra 487 (2017) 173-199], it was shown that the ideals of L satisfy the distributive law, a property of Prufer 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 ufer domains which are the ideal versions of two theorems in Elementary Number Theory, namely, for positive integers a, b, c, gcd(a, b).lcm(a, b) = a.b and a.gcd(b, c) = 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 nonzero ideals.