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PreprintCC BY-NC-ND 4.0arXiv

A note on the ideal lattice of a weighted Leavitt path algebra

  • Raimund Preusser
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Abstract

Let K be a field and LK(E,w) the Leavitt path algebra of a row-finite weighted graph (E,w). Suppose that LK(E,w) is not isomorphic to an ordinary Leavitt path algebra. We prove that the ideal lattice of every countably generated unital K-algebra embeds as a closed interval into the ideal lattice of LK(E,w). The embedding preserves multiplication of ideals modulo the lower endpoint of the interval. It follows that the ideal lattice of LK(E,w) is not distributive and that ideal multiplication in LK(E,w) is not commutative. Moreover, we show that if (E,w) is countable, then LK(E,w) has exactly |K|ℵ0 ideals, prime ideals, left primitive ideals, and right primitive ideals.

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Cite

BibTeXRIS
BibTeX
@misc{preusser2026note,
  title = {{A note on the ideal lattice of a weighted Leavitt path algebra}},
  author = {Raimund Preusser},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39842},
  eprint = {2610.39842},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39842},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - A note on the ideal lattice of a weighted Leavitt path algebra
AU  - Raimund Preusser
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39842
UR  - https://arxiv.org/abs/2610.39842
AB  - Let K be a field and L_K(E,w) the Leavitt path algebra of a row-finite weighted graph (E,w). Suppose that L_K(E,w) is not isomorphic to an ordinary Leavitt path algebra. We prove that the ideal lattice of every countably generated unital K-algebra embeds as a closed interval into the ideal lattice of L_K(E,w). The embedding preserves multiplication of ideals modulo the lower endpoint of the interval. It follows that the ideal lattice of L_K(E,w) is not distributive and that ideal multiplication in L_K(E,w) is not commutative. Moreover, we show that if (E,w) is countable, then L_K(E,w) has exactly |K|^ℵ_0 ideals, prime ideals, left primitive ideals, and right primitive ideals.
ER  -