Certain strongly clean matrices over local rings

Authors : Tuğçe Pekacar ÇALCI, Huanyin CHEN

Pages : 2296-2303

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Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :We are concerned about various strongly clean properties of a kind of matrix subrings $L_{insert ignore into journalissuearticles values(s);}insert ignore into journalissuearticles values(R);$ over a local ring $R$. Let $R$ be a local ring, and let $s\in Cinsert ignore into journalissuearticles values(R);$. We prove that $A\in L_{insert ignore into journalissuearticles values(s);}insert ignore into journalissuearticles values(R);$ is strongly clean if and only if $A$ or $I_2-A$ is invertible, or $A$ is similar to a diagonal matrix in $L_{insert ignore into journalissuearticles values(s);}insert ignore into journalissuearticles values(R);$. Furthermore, we prove that $A\in L_{insert ignore into journalissuearticles values(s);}insert ignore into journalissuearticles values(R);$ is quasipolar if and only if $A\in GL_2insert ignore into journalissuearticles values(R);$ or $A\in L_{insert ignore into journalissuearticles values(s);}insert ignore into journalissuearticles values(R);^{qnil}$, or $A$ is similar to a diagonal matrix $\leftinsert ignore into journalissuearticles values( \begin{array}{cc} \lambda&0\\ 0&\mu \end{array} \right);$ in $L_{insert ignore into journalissuearticles values(s);}insert ignore into journalissuearticles values(R);$, where $\lambda\in Jinsert ignore into journalissuearticles values(R);$, $\mu\in Uinsert ignore into journalissuearticles values(R);$ or $\lambda\in Uinsert ignore into journalissuearticles values(R);$, $\mu\in Jinsert ignore into journalissuearticles values(R);$, and $l_{\mu}-r_{\lambda}$, $l_{\lambda}-r_{\mu}$ are injective. Pseudopolarity of such matrix subrings is also obtained.
Keywords : Matrix ring, strongly clean matrix, quasipolar matrix