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1. CMB 2009 (vol 52 pp. 145)
$2$-Clean Rings A ring $R$ is said to be $n$-clean if every
element can be written as a sum of an idempotent and $n$ units.
The class of these rings contains clean rings and $n$-good rings
in which each element is a sum of $n$ units. In this paper, we
show that for any ring $R$, the endomorphism ring of a free
$R$-module of rank at least 2 is $2$-clean and that the ring $B(R)$
of all $\omega\times \omega$ row and column-finite matrices over
any ring $R$ is $2$-clean. Finally, the group ring $RC_{n}$ is
considered where $R$ is a local ring.
Keywords:$2$-clean rings, $2$-good rings, free modules, row and column-finite matrix rings, group rings Categories:16D70, 16D40, 16S50 |
2. CMB 2006 (vol 49 pp. 265)
Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial 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 $R$.
Keywords:Clean rings, linear transformations, endomorphism rings Categories:16S50, 16E50 |