A Theorem on Unit-Regular Rings

Lee, Tsiu-Kwen ; Zhou, Yiqiang

Canadian mathematical bulletin, Tome 53 (2010) no. 2, pp. 321-326

Résumé

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 $n\,\ge \,0$ .

Détail

DOI : 10.4153/CMB-2010-023-0

Mots-clés : 16E50, 16U99, 16S70, 16S35, morphic rings, unit-regular rings, skew polynomial rings