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

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${R}/{J(R)}=2$.
Nil-clean matrix, Idempotent matrix, Nilpotent matrix, Principal ideal domain
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Subjects Engineering Articles Author: Somayeh Hadjirezaei Author: Somayeh Karimzadeh Publication Date : May 15, 2016
 Bibtex @research article { jacodesmath285390, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2016}, volume = {3}, pages = {91 - 96}, doi = {10.13069/jacodesmath.82415}, title = {Matrix rings over a principal ideal domain in which elements are nil-clean}, key = {cite}, author = {Hadjirezaei, Somayeh and Karimzadeh, Somayeh} } APA Hadjirezaei, S , Karimzadeh, S . (2016). Matrix rings over a principal ideal domain in which elements are nil-clean . Journal of Algebra Combinatorics Discrete Structures and Applications , 3 (2) , 91-96 . DOI: 10.13069/jacodesmath.82415 MLA Hadjirezaei, S , Karimzadeh, S . "Matrix rings over a principal ideal domain in which elements are nil-clean" . Journal of Algebra Combinatorics Discrete Structures and Applications 3 (2016 ): 91-96 Chicago Hadjirezaei, S , Karimzadeh, S . "Matrix rings over a principal ideal domain in which elements are nil-clean". Journal of Algebra Combinatorics Discrete Structures and Applications 3 (2016 ): 91-96 RIS TY - JOUR T1 - Matrix rings over a principal ideal domain in which elements are nil-clean AU - Somayeh Hadjirezaei , Somayeh Karimzadeh Y1 - 2016 PY - 2016 N1 - doi: 10.13069/jacodesmath.82415 DO - 10.13069/jacodesmath.82415 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 91 EP - 96 VL - 3 IS - 2 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.82415 UR - https://doi.org/10.13069/jacodesmath.82415 Y2 - 2020 ER - EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications Matrix rings over a principal ideal domain in which elements are nil-clean %A Somayeh Hadjirezaei , Somayeh Karimzadeh %T Matrix rings over a principal ideal domain in which elements are nil-clean %D 2016 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 3 %N 2 %R doi: 10.13069/jacodesmath.82415 %U 10.13069/jacodesmath.82415 ISNAD Hadjirezaei, Somayeh , Karimzadeh, Somayeh . "Matrix rings over a principal ideal domain in which elements are nil-clean". Journal of Algebra Combinatorics Discrete Structures and Applications 3 / 2 (May 2016): 91-96 . https://doi.org/10.13069/jacodesmath.82415 AMA Hadjirezaei S , Karimzadeh S . Matrix rings over a principal ideal domain in which elements are nil-clean. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016; 3(2): 91-96. Vancouver Hadjirezaei S , Karimzadeh S . Matrix rings over a principal ideal domain in which elements are nil-clean. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016; 3(2): 91-96.

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