Matrix rings over a principal ideal domain in which elements are nil-clean

Year 2016, Volume: 3 Issue: 2, 91 - 96, 15.05.2016

Somayeh Hadjirezaei Somayeh Karimzadeh

https://doi.org/10.13069/jacodesmath.82415

Abstract

An element of a ring $R$ is called nil-clean if it is the sum of an idempotent and a nilpotent element. A ring is called nil-clean if each of its elements is nil-clean. S. Breaz et al. in \cite{Bre} proved their main result that the matrix ring $\mathbb{M}_{ n}(F)$ over a field $F$ is nil-clean if and only if $F\cong \mathbb{F}_2$, where $\mathbb{F}_2$ is the field of two elements. M. T. Ko\c{s}an et al. generalized this result to a division ring. In this paper, we show that the $n\times n$ matrix ring over a principal ideal domain $R$ is a nil-clean ring if and only if $R$ is isomorphic to $\mathbb{F}_2$. Also, we show that the same result is true for the $2\times 2$ matrix ring over an integral domain $R$. As a consequence, we show that for a commutative ring $R$, if $\mathbb{M}_{ 2}(R)$ is a nil-clean ring, then dim$R=0$ and char${R}/{J(R)}=2$.

Keywords

Nil-clean matrix, Idempotent matrix, Nilpotent matrix, Principal ideal domain

References

• [1] S. Breaz, G. Calugareanu, P. Danchev, T. Micu, Nil-clean matrix rings, Linear Algebra Appl. 439(10) (2013) 3115-3119.
• [2] J. Chen, X. Yang, Y. Zhou, On strongly clean matrix and triangular matrix rings, Comm. Algebra. 34(10) (2006) 3659–3674.
• [3] A. J. Diesl, Classes of strongly clean rings, Ph. D. thesis, University of California, Berkeley, 2006.
• [4] A. J. Diesl, Nil clean rings, J. Algebra. 383 (2013) 197–211.
• [5] T. W. Hungerford, Algebra, Springer-Verlag, 1980.
• [6] M.T. Kosan, T. K. Lee, Y. Zhou, When is every matrix over a division ring a sum of an idempotent and a nilpotent?, Linear Algebra Appl. 450 (2014) 7–12.
• [7] T. Kosan, Z. Wang, Y. Zhou, Nil-clean and strongly nil-clean rings, J. Pure Appl. Algebra. 220(2) (2016) 633–646.
• [8] W. K. Nicholson, Strongly clean rings and Fitting’s lemma, Comm. Algebra. 27(8) (1999) 3583–3592.
• [9] G. Song, X. Guo, Diagonability of idempotent matrices over noncommutative rings, Linear Algebra Appl. 297(1-3) (1999) 1–7.