The nil-clean $2\times 2$ integral units

Year 2021, Volume: 50 Issue: 1, 41 - 45, 04.02.2021

Grigore Calugareanu

https://doi.org/10.15672/hujms.622655

Abstract

We prove that all trace $1$, $2\times 2$ invertible matrices over $\mathbb{Z}$ are nil-clean and, up to similarity, that there are only two trace $1$, $2\times 2$ invertible matrices over $\mathbb{Z}$.

Keywords

nil-clean, clean, similarity, binary quadratic form, class number

References

• [1] T. Andreescu and D. Andrica, Number Theory. Structures, examples, and problems, Birkhäuser Boston, Inc., Boston, MA, 2009.
• [2] A.I. Borevich and I.R. Shafarevich, Number Theory. Pure and Applied Mathematics Vol. 20, Academic Press, New York-London, 1966.
• [3] J. Buchmann and U. Vollmer, Binary Quadratic Forms. Springer, Berlin 2007.
• [4] A.J. Diesl, Nil clean rings, J. Algebra 383, 197-211, 2013.
• [5] W.K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229, 269-278, 1977.
• [6] A. Steger, Diagonalibility of Idempotent Matrices. Pacific J. Math. 19 (3), 535-542, 1966.