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A Theorem on Unit-Regular Rings
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- Journal:
- Canadian Mathematical Bulletin / Volume 53 / Issue 2 / 01 June 2010
- Published online by Cambridge University Press:
- 20 November 2018, pp. 321-326
- Print publication:
- 01 June 2010
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- Article
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- You have access
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Let
$R$ be a unit-regular ring and let 
$\sigma $ be an endomorphism of $R$ such that 
$\sigma \left( e \right)\,=\,e$ for all 
${{e}^{2}}\,=\,e\,\in \,R$ and let 
$n\,\ge \,0$. It is proved that every element of 
$R[x;\,\sigma ]/\left( {{x}^{n+1}} \right)$ is equivalent to an element of the form 
${{e}_{0}}\,+\,{{e}_{1}}x\,+\,\cdots \,+\,{{e}_{n}}{{x}^{n}}$, where the 
${{e}_{i}}$ are orthogonal idempotents of $R$. As an application, it is proved that $R[x;\,\sigma ]/\left( {{x}^{n+1}} \right)$ is left morphic for each 
$n\,\ge \,0$.
