On Pr\'ufer-Like Properties of Leavitt Path Algebras
| Autor: | Esin, Songül, Kanuni, Müge, Koç, Ayten, Radler, Katherine, Rangaswamy, Kulumani M. |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | J Algebra Appl- Vol.19 No. 07, 2050122 (2020) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1142/S0219498820501224 |
| Popis: | 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. Comment: 18 pages |
| Databáze: | arXiv |
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