Geodesic
Decompositions of $2\times 2$ Matrices Over Local Rings
Publications de l'Institut Mathématique, _N_S_100 (2016) no. 114, p. 287

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An element $a$ of a ring $R$ is called perfectly clean if there exists an idempotent $e\in\comm^2(a)$ such that $a-e\in U(R)$. A ring $R$ is perfectly clean in case every element in $R$ is perfectly clean. In this paper, we completely determine when every $2\times 2$ matrix and triangular matrix over local rings are perfectly clean. These give more explicit characterizations of strongly clean matrices over local rings. We also obtain several criteria for a triangular matrix to be perfectly J-clean. For instance, it is proved that for a commutative local ring $R$, every triangular matrix is perfectly J-clean in $T_n(R)$ if and only if $R$ is strongly J-clean.
DOI : 10.2298/PIM1614287C
Classification : 16S50, 16S70, 16U99
Keywords: perfectly clean ring, perfectly J-clean ring, quasipolar ring, matrix, triangular matrix 
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     author = {Huanyin Chen and Sait Halicioglu and Handan Kose},
     title = {Decompositions of $2\times 2$ {Matrices} {Over} {Local} {Rings}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {287 },
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     year = {2016},
     doi = {10.2298/PIM1614287C},
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     url = {http://geodesic.mathdoc.fr/articles/10.2298/PIM1614287C/}
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Huanyin Chen; Sait Halicioglu; Handan Kose. Decompositions of $2\times 2$ Matrices Over Local Rings. Publications de l'Institut Mathématique, _N_S_100 (2016) no. 114, p. 287 . doi: 10.2298/PIM1614287C

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