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On Prüfer-like properties of Leavitt path algebras.
- Source :
-
Journal of Algebra & Its Applications . Jul2020, Vol. 19 Issue 7, pN.PAG-N.PAG. 19p. - Publication Year :
- 2020
-
Abstract
- 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 (a b , a c). 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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *ALGEBRA
*PRIME ideals
*NUMBER theory
Subjects
Details
- Language :
- English
- ISSN :
- 02194988
- Volume :
- 19
- Issue :
- 7
- Database :
- Academic Search Index
- Journal :
- Journal of Algebra & Its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 144828694
- Full Text :
- https://doi.org/10.1142/S0219498820501224