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Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial
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- Journal:
- Canadian Mathematical Bulletin / Volume 49 / Issue 2 / 01 June 2006
- Published online by Cambridge University Press:
- 20 November 2018, pp. 265-269
- Print publication:
- 01 June 2006
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- Article
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If
$C=C\left( R \right)$ denotes the center of a ring 
$R$ and 
$g\left( x \right)$ is a polynomial in 
$C\left[ x \right]$, Camillo and Simón called a ring $g\left( x \right)$-clean if every element is the sum of a unit and a root of $g\left( x \right)$. If 
$V$ is a vector space of countable dimension over a division ring 
$D$, they showed that 
$\text{en}{{\text{d}}_{\,D}}V$ is $g\left( x \right)$-clean provided that $g\left( x \right)$ has two roots in 
$C\left( D \right)$. If 
$g\left( x \right)=x-{{x}^{2}}$ this shows that $\text{en}{{\text{d}}_{\,D}}V$ is clean, a result of Nicholson and Varadarajan. In this paper we remove the countable condition, and in fact prove that 
$\text{en}{{\text{d}}_{\,R}}M$ is $g\left( x \right)$-clean for any semisimple module 
$M$ over an arbitrary ring 
$R$ provided that 
$g\left( x \right)\in \left( x-a \right)\left( x-b \right)C\left[ x \right] $ where 
$a,b\in C$ and both 
$b$ and 
$b-a$ are units in $R$.
