Jacobsons Lemma fails for nil-clean 2 × 2 integral matrices

Călugăreanu, Grigore; Pop, Horia F.

doi:10.2478/auom-2020-0035

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Jacobsons Lemma fails for nil-clean 2 × 2 integral matrices

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 28 (2020): Issue 3 (December 2020)

By: Grigore Călugăreanu and Horia F. Pop

Open Access

|Dec 2020

Abstract

We show that for two 2 × 2 integral matrices A, B, if the product AB is nil-clean then BA may not be nil-clean. Despite the fact that for many special cases, BA is also nil-clean, we finally found three counterexamples. All the way, the computer aid was decisive.

References

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[2] G. Călugăreanu The nil-clean 2 × 2 integral units. (6 pages) submitted.
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[3] G. Călugăreanu Nil-clean integral 2 × 2 matrices: The elliptic case. Bull. Math. Soc. Sci. Math. Roumanie 62 (110) (3), 2019, 239-250.
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[4] K. Matthews Solving the general quadratic Diophantine equation ax2 + bxy + cy2 + dx + ey + f = 0. http://www.numbertheory.org/php/generalquadratic.html
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DOI: https://doi.org/10.2478/auom-2020-0035 | Journal eISSN: 1844-0835 | Journal ISSN: 1224-1784

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Language: English

Page range: 83 - 88

Submitted on: Jan 2, 2020

Accepted on: Jan 17, 2020

Published on: Dec 28, 2020

Published by: Ovidius University of Constanta

In partnership with: Paradigm Publishing Services

Publication frequency: 3 issues per year

Keywords:

nil-clean,

clean,

2 × 2 matrix,

principal ideal domain

Related subjects:

Mathematics,

General mathematics

© 2020 Grigore Călugăreanu, Horia F. Pop, published by Ovidius University of Constanta
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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