Topologically boolean and $g\left(x\right)$-clean rings

Angelina Yan Mui Chin, Kiat Tat Qua

Abstract: Let $R$ be a ring with identity and let $g\left(x\right)$ be a polynomial in $Z\left(R\right)\left[x\right]$ where $Z\left(R\right)$ denotes the center of $R$. An element $r\in R$ is called $g\left(x\right)$-clean if $r=u+s$ for some $u,s\in R$ such that $u$ is a unit and $g\left(s\right)=0$. The ring $R$ is $g\left(x\right)$-clean if every element of $R$ is $g\left(x\right)$-clean. We consider $g\left(x\right)=x(x-c)$ where $c$ is a unit in $R$ such that every root of $g\left(x\right)$ is central in $R$. We show, via set-theoretic topology, that among conditions equivalent to $R$ being $g\left(x\right)$-clean, is that $R$ is right (left) $c$-topologically boolean.

Keywords: $g\left(x\right)$-clean, $n$-clean, topologically boolean

Classification (MSC2000): 16U99

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