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On Prüfer-like properties of Leavitt path algebras
(World Scientific Publishing Co. Pte Ltd, 2019) Esin, Songül; Er, Müge Kanuni; Koç, Ayten; Radler, Katherine; Rangaswamy, Kulumani M.
Prüfer 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 Prüfer 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 Prüfer 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. © 2020 World Scientific Publishing Company.