scholarly journals On Prüfer-like properties of Leavitt path algebras

2019 ◽  
Vol 19 (07) ◽  
pp. 2050122 ◽  
Author(s):  
Songül Esin ◽  
Müge Kanuni ◽  
Ayten Koç ◽  
Katherine Radler ◽  
Kulumani M. Rangaswamy
Keyword(s):  
Sufficient Conditions ◽  
Ideal Theory ◽  
Prime Ideals ◽  
Integral Domains ◽  
Ideal Lattices ◽  
Path Algebras ◽  
Dedekind Domains ◽  
Prufer Domains ◽  
Leavitt Path Algebras ◽  
Prüfer Domains

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 [Formula: see text], 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 [Formula: see text] satisfy the distributive law, a property of Prüfer domains and that [Formula: see text] is a multiplication ring, a property of Dedekind domains. In this paper, we first show that [Formula: see text] satisfies two more characterizing properties of Prüfer domains which are the ideal versions of two theorems in Elementary Number Theory, namely, for positive integers [Formula: see text], [Formula: see text] and [Formula: see text]. We also show that [Formula: see text] 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 [Formula: see text] satisfies another important characterizing property of almost Dedekind domains, namely, the cancellative property of its nonzero ideals.

scholarly journals Products and intersections of prime-power ideals in Leavitt path algebras

2021 ◽  
pp. 2250104
Author(s):  
Zachary Mesyan ◽  
Kulumani M. Rangaswamy
Keyword(s):  
Prime Power ◽  
Path Algebra ◽  
Ideal Theory ◽  
Prime Ideals ◽  
Leavitt Path Algebra ◽  
Path Algebras ◽  
Leavitt Path Algebras

We continue a very fruitful line of inquiry into the multiplicative ideal theory of an arbitrary Leavitt path algebra [Formula: see text]. Specifically, we show that factorizations of an ideal in [Formula: see text] into irredundant products or intersections of finitely many prime-power ideals are unique, provided that the ideals involved are powers of distinct prime ideals. We also characterize the completely irreducible ideals in [Formula: see text], which turn out to be prime-power ideals of a special type, as well as ideals that can be factored into products or intersections of finitely many completely irreducible ideals.

scholarly journals Decomposition and classification of length functions

2020 ◽  
Vol 32 (5) ◽  
pp. 1109-1129
Author(s):  
Dario Spirito
Keyword(s):  
Integral Domain ◽  
Integral Domains ◽  
Bijective Correspondence ◽  
Length Functions ◽  
Standard Representation ◽  
Prufer Domains ◽  
Prüfer Domains

AbstractWe study decompositions of length functions on integral domains as sums of length functions constructed from overrings. We find a standard representation when the integral domain admits a Jaffard family, when it is Noetherian and when it is a Prüfer domains such that every ideal has only finitely many minimal primes. We also show that there is a natural bijective correspondence between singular length functions and localizing systems.

On ideal theory in high prufer domains

1975 ◽  
Vol 14 (4) ◽  
pp. 303-336 ◽  
Author(s):  
Moshe Jarden
Keyword(s):  
Ideal Theory ◽  
Prufer Domains ◽  
Prüfer Domains

scholarly journals INTEGRAL DOMAINS WHOSE SIMPLE OVERRINGS ARE INTERSECTIONS OF LOCALIZATIONS

2005 ◽  
Vol 04 (02) ◽  
pp. 195-209 ◽  
Author(s):  
MARCO FONTANA ◽  
EVAN HOUSTON ◽  
THOMAS LUCAS
Keyword(s):  
Dedekind Domain ◽  
Quotient Field ◽  
Integral Domains ◽  
Finiteness Conditions ◽  
Integrally Closed ◽  
Prufer Domains ◽  
Prüfer Domains

Call a domain R an sQQR-domain if each simple overring of R, i.e., each ring of the form R[u] with u in the quotient field of R, is an intersection of localizations of R. We characterize Prüfer domains as integrally closed sQQR-domains. In the presence of certain finiteness conditions, we show that the sQQR-property is very strong; for instance, a Mori sQQR-domain must be a Dedekind domain. We also show how to construct sQQR-domains which have (non-simple) overrings which are not intersections of localizations.

scholarly journals Full idempotents in Leavitt path algebras

2019 ◽  
Vol 18 (04) ◽  
pp. 1950062
Author(s):  
Ekrem Emre
Keyword(s):  
Directed Graph ◽  
Sufficient Conditions ◽  
Path Algebra ◽  
Necessary And Sufficient Conditions ◽  
Leavitt Path Algebra ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Necessary And Sufficient

We give necessary and sufficient conditions on a directed graph [Formula: see text] for which the associated Leavit path algebra [Formula: see text] has at least one full idempotent. Also, we define [Formula: see text] sub-graphs of [Formula: see text] and show that [Formula: see text] has at least one full idempotent if and only if there is a sub-graph [Formula: see text] such that the associated Leavitt path algebra [Formula: see text] has at least one full idempotent.

On graded primitive leavitt path algebras

2020 ◽  
pp. 2150173
Author(s):  
Kulumani M. Rangaswamy
Keyword(s):  
Directed Graphs ◽  
Sufficient Conditions ◽  
Path Algebra ◽  
Prime Rings ◽  
Leavitt Path Algebra ◽  
Von Neumann ◽  
Path Algebras ◽  
Von Neumann Regular ◽  
Leavitt Path Algebras ◽  
Necessary And Sufficient

Graded primitive Leavitt path algebras of arbitrary directed graphs over a field [Formula: see text] are completely characterized by means of graphical conditions. Necessary and sufficient conditions are given under which a graded prime Leavitt path algebra becomes graded primitive and this leads to answering the graded version of a question of Kaplansky on von Neumann regular prime rings in the context of Leavitt path algebras.

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