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PreprintCC BY-NC-ND 4.0arXiv

Hopkins-Levitzki Type Theorems for Groupoid Graded Rings

  • Zaqueu Cristiano
  • Wellington Marques de Souza
  • Javier Sánchez
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Abstract

We continue the study of the basic theory of object-unital groupoid graded rings. In this work, we are especially interested in nilpotency conditions on the graded Jacobson radical. We introduce the concept of left/right objectwise nilpotency of graded ideals, and prove that this condition is appropriate for obtaining graded generalizations of the Hopkins--Levitzki theorem. Although this condition is not symmetric, we show that its two-sided version is suitable for defining gr-semiprimary rings. It is known that one-sided Γ0-artinian rings need not be Γ0-noetherian, but using our tools we prove that one-sided gr-hereditary Γ0-artinian rings, two-sided Γ0-artinian rings, and d-finitely generated one-sided Γ0-artinian rings are Γ0-noetherian. However, the first class need not be gr-semiprimary, whereas the other two always are. We illustrate our results with several (counter)examples, especially involving graded upper triangular matrices.

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Cite

BibTeXRIS
BibTeX
@misc{cristiano2026hopkinslevitzki,
  title = {{Hopkins-Levitzki Type Theorems for Groupoid Graded Rings}},
  author = {Zaqueu Cristiano and Wellington Marques de Souza and Javier Sánchez},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00305},
  eprint = {2609.00305},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00305},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Hopkins-Levitzki Type Theorems for Groupoid Graded Rings
AU  - Zaqueu Cristiano
AU  - Wellington Marques de Souza
AU  - Javier Sánchez
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00305
UR  - https://arxiv.org/abs/2609.00305
AB  - We continue the study of the basic theory of object-unital groupoid graded rings. In this work, we are especially interested in nilpotency conditions on the graded Jacobson radical. We introduce the concept of left/right objectwise nilpotency of graded ideals, and prove that this condition is appropriate for obtaining graded generalizations of the Hopkins--Levitzki theorem. Although this condition is not symmetric, we show that its two-sided version is suitable for defining gr-semiprimary rings. It is known that one-sided Γ_0-artinian rings need not be Γ_0-noetherian, but using our tools we prove that one-sided gr-hereditary Γ_0-artinian rings, two-sided Γ_0-artinian rings, and d-finitely generated one-sided Γ_0-artinian rings are Γ_0-noetherian. However, the first class need not be gr-semiprimary, whereas the other two always are. We illustrate our results with several (counter)examples, especially involving graded upper triangular matrices.
ER  -