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Search: MSC category 16U99 ( None of the above, but in this section )

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1. CMB Online first

Ara, Pere; O'Meara, Kevin C.
The Nilpotent Regular Element Problem
We use George Bergman's recent normal form for universally adjoining an inner inverse to show that, for general rings, a nilpotent regular element $x$ need not be unit-regular. This contrasts sharply with the situation for nilpotent regular elements in exchange rings (a large class of rings), and for general rings when all powers of the nilpotent element $x$ are regular.

Keywords:nilpotent element, von Neumann regular element, unit-regular, Bergman's normal form
Categories:16E50, 16U99, 16S10, 16S15

2. CMB 2009 (vol 53 pp. 321)

Lee, Tsiu-Kwen; Zhou, Yiqiang
A Theorem on Unit-Regular Rings
Let $R$ be a unit-regular ring and let $\sigma $ be an endomorphism of $R$ such that $\sigma (e)=e$ for all $e^2=e\in R$ and let $n\ge 0$. It is proved that every element of $R[x \mathinner;\sigma]/(x^{n+1})$ is equivalent to an element of the form $e_0+e_1x+\dots +e_nx^n$, where the $e_i$ are orthogonal idempotents of $R$. As an application, it is proved that $R[x \mathinner; \sigma ]/(x^{n+1})$ is left morphic for each $n\ge 0$.

Keywords:morphic rings, unit-regular rings, skew polynomial rings
Categories:16E50, 16U99, 16S70, 16S35

3. CMB 1999 (vol 42 pp. 174)

Ferrero, Miguel; Sant'Ana, Alveri
Rings With Comparability
The class of rings studied in this paper properly contains the class of right distributive rings which have at least one completely prime ideal in the Jacobson radical. Amongst other results we study prime and semiprime ideals, right noetherian rings with comparability and prove a structure theorem for rings with comparability. Several examples are also given.

Categories:16U99, 16P40, 16D14, 16N60

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