Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial

http://dx.doi.org/10.4153/CMB-2006-027-6
Canad. Math. Bull. 49(2006), 265-269

  Published:2006-06-01
 Printed: Jun 2006

If $C=C(R)$ denotes the center of a ring $R$ and $g(x)$ is a polynomial in C[x]$, Camillo and Sim\'{o}n called a ring $g(x)$-clean if every element is the sum of a unit and a root of $g(x)$. If $V$ is a vector space of countable dimension over a division ring $D,$ they showed that $\end {}_{D}V$ is $g(x)$-clean provided that $g(x)$ has two roots in $C(D)$. If $g(x)=x-x^{2}$ this shows that $\end {}_{D}V$ is clean, a result of Nicholson and Varadarajan. In this paper we remove the countable condition, and in fact prove that $\Mend {}_{R}M$ is $g(x)$-clean for any semisimple module $M$ over an arbitrary ring $R$ provided that $g(x)\in (x-a)(x-b)C[x]$ where $a,b\in C$ and both $b$ and $b-a$ are units in $R$.