On Prüfer-like properties of Leavitt path algebras

Esin, S.K.; Kanuni̇, Müge; Koç, Ayten; Radler, Katherine; Rangaswamy, Kulumani M.

doi:10.1142/s0219498820501224

Cited by 6 publications

(6 citation statements)

References 9 publications

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“…While conditions (2)- (5) in the above definition are generally not interchangeable, they happen to coincide in Leavitt path algebras. Proposition 3.2.…”

Section: Multiplicative Conditions On Idealsmentioning

confidence: 99%

Products of ideals in Leavitt path algebras

Abrams

Mesyan

Rangaswamy

2020

Communications in Algebra

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Ideals in Leavitt path algebras have been shown to share many properties with those of integral domains. Since studying factorizations of ideals in integral domains into special types of ideals (particularly, prime, prime-power, primary, irreducible, semiprime, and quasi-primary ones) has proved fruitful, we conduct an analogous investigation in the context of Leavitt path algebras. Specifically, we classify the proper ideals in these rings that admit factorizations into products of each of the above types of ideals. We also classify the Leavitt path algebras where every proper ideal admits a factorization of each of these sorts, as well as those Leavitt path algebras where every proper ideal is of one of those types.

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“…While conditions (2)- (5) in the above definition are generally not interchangeable, they happen to coincide in Leavitt path algebras. Proposition 3.2.…”

Section: Multiplicative Conditions On Idealsmentioning

confidence: 99%

Products of ideals in Leavitt path algebras

Abrams

Mesyan

Rangaswamy

2020

Communications in Algebra

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“…Upon dropping terms, if needed, we may assume that the intersection is irredundant. If I is non-graded, then, upon further reindexing, Lemma 4.7 (2,4,5) gives that |Y | = n ≤ m for some positive integer n, that for each i ∈ {1, . .…”

Section: Products and Intersections Of Completely Irreducible Idealsmentioning

confidence: 99%

“…In particular, a number of papers have been devoted to characterizing special types of ideals of L in terms of graphical properties of E, and to describing the ideals of L that can be factored into products of ideals of these types. More specifically prime, primitive, semiprime, and irreducible ideals have received such treatment in the literature-see [2,3,4,5,12,14]. An interesting feature of Leavitt path algebras is that, while they are highly noncommutative, multiplication of their ideals is commutative, and further, their ideals share a number of properties with ideals in various commutative rings, such as Dedekind domains (where ideals are projective), Bézout rings (where finitely-generated ideals are principal), arithmetical rings (where the ideal lattices are distributive), and Prüfer domains (where the ideal lattices have special properties).…”

Section: Introductionmentioning

confidence: 99%

Products and Intersections of Prime-Power Ideals in Leavitt Path Algebras

Mesyan¹,

Rangaswamy²

2021

Preprint

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We continue a very fruitful line of inquiry into the multiplicative ideal theory of an arbitrary Leavitt path algebra L. Specifically, we show that factorizations of an ideal in L 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 L, which turn out to be primepower ideals of a special type, as well as ideals that can be factored into products or intersections of finitely many completely irreducible ideals.

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“…In a recent paper [7], it has been shown that the ideals of a Leavitt path algebra satisfy two more characterizing properties of Prüfer domains among integral domains.…”

Section: The Lattice Of Ideals Of a Leavitt Path Algebramentioning

confidence: 99%

The multiplicative ideal theory of Leavitt path algebras

Rangaswamy

2017

Journal of Algebra

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It is shown that every Leavitt path algebra L of an arbitrary directed graph E over a field K is an arithmetical ring, that is, the distributive law A ∩ (B + C) = (A ∩ B) + (A ∩ C) holds for any three two-sided ideals of L. It is also shown that L is a multiplication ring, that is, given any two ideals A, B in L with A ⊆ B, there is always an ideal C such that A = BC, an indication of a possible rich multiplicative ideal theory for L. Existence and uniqueness of factorization of the ideals of L as products of special types of ideals such as prime, irreducible or primary ideals is investigated. The irreducible ideals of L turn out to be precisely the primary ideals of L. It is shown that an ideal I of L is a product of finitely many prime ideals if and only the graded part gr(I) of I is a product of prime ideals and I/gr(I) is finitely generated with a generating set of cardinality no more than the number of distinct prime ideals in the prime factorization of gr(I). As an application, it is shown that if E is a finite graph, then every ideal of L is a product of prime ideals. The same conclusion holds if L is two-sided artinian or two-sided noetherian. Examples are constructed verifying whether some of the well-known theorems in the ideal theory of commutative rings such as the Cohen's theorem on prime ideals and the characterizing theorem on ZPI rings hold for Leavitt path algebras.

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On Prüfer-like properties of Leavitt path algebras

Cited by 6 publications

References 9 publications

Products of ideals in Leavitt path algebras

Products of ideals in Leavitt path algebras

Products and Intersections of Prime-Power Ideals in Leavitt Path Algebras

The multiplicative ideal theory of Leavitt path algebras

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