A classification of ideals in Steinberg and Leavitt path algebras over arbitrary rings

Rigby, Simon W.; Hove, Thibaud van den

doi:10.1016/j.jalgebra.2021.08.021

Cited by 2 publications

(3 citation statements)

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“…Larki [8] made a study for ideals with coefficients in a commutative ring. A recent paper by Rigby and van den Hove [10] about generators of ideals in Leavitt path algebras over a commutative ring R with identity, proves that a two-sided ideal of a Leavitt path algebra L R (E) is generated by elements of the following three types:…”

Section: (Thementioning

confidence: 99%

On nil ideals of Leavitt path algebras over commutative rings

Dutta,

Buhphang

2024

Discuss. Math. - General Algebra and Applications

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We show in this paper that for any graph E and for a commutative unital ring R, the nil ideals of the Leavitt path algebra L R (E) depend solely on the nil ideals of the ring R. A connection between the Jacobson radical of L R (E) and the Jacobson radical of R is obtained. We also prove that for a nil ideal I of a Leavitt path algebra L R (E) the ideal M 2 (I) is also nil, thus obtaining that Leavitt path algebras over arbitrary graphs satisfy the Köethe's conjecture.

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Section: (Thementioning

confidence: 99%

On nil ideals of Leavitt path algebras over commutative rings

Dutta,

Buhphang

2024

Discuss. Math. - General Algebra and Applications

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“…Part of our aim is to provide a more general description of ideals in Leavitt path algebras over a ring than in [6]. We were not the only ones interested in such results; in a paper [16] published last December, Rigby and van den Hove give a complete description of the ideal lattice of Leavitt path algebras over arbitrary rings. Since Rigby and van den Hove focus on classifying the ideals of any Leavitt path algebra over an arbitrary ring, the lattice they use to do this is complicated and the join is not straightforward [16,Proposition 6.16.].…”

Section: Chapter 1 Introductionmentioning

confidence: 99%

“…We were not the only ones interested in such results; in a paper [16] published last December, Rigby and van den Hove give a complete description of the ideal lattice of Leavitt path algebras over arbitrary rings. Since Rigby and van den Hove focus on classifying the ideals of any Leavitt path algebra over an arbitrary ring, the lattice they use to do this is complicated and the join is not straightforward [16,Proposition 6.16.]. Our results provide a lattice whose join and meet operations replicate those of an ideal lattice, thus our lattice is simpler and has a more intuitive join.…”

Section: Chapter 1 Introductionmentioning

confidence: 99%

Lattices of ideals in graph algebras: refining the join operation

McDougall¹

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<p>We consider the ideal structure of Leavitt path algebras of graphs that satisfy Condition (K) over commutative rings with identity. The approach we use takes advantage of the relationship between Leavitt path algebras and Steinberg algebras of boundary path groupoids. We establish a lattice $\mathcal{F}'$ consisting of particular maps $\tau:E^0\to \LR$ and show that this lattice is isomorphic to the lattice of ideals of $L_R(E)$. The advantage to our approach over previous lattice isomorphisms, even in the case that $R$ is a field, is that we obtain convenient join and meet operations in $\mathcal{F}'$. Lastly, we provide three concrete examples.</p>

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A classification of ideals in Steinberg and Leavitt path algebras over arbitrary rings

Cited by 2 publications

References 27 publications

On nil ideals of Leavitt path algebras over commutative rings

On nil ideals of Leavitt path algebras over commutative rings

Lattices of ideals in graph algebras: refining the join operation

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