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On Abelian \(\pi\)-regular rings. (English) Zbl 0881.16003

A \(\pi\)-regular ring \(R\) is said to be abelian \(\pi\)-regular if all its idempotents are central. Let \(\text{Nil}(R)\) denote the set of all nilpotent elements of \(R\). The author proves that a ring \(R\) whose idempotents are central is \(\pi\)-regular if and only if \(\text{Nil}(R)\) is a two-sided ideal of \(R\) and \(R/\text{Nil}(R)\) is regular. The necessary part of this result was earlier shown by Ohori, but the author here provides an alternative proof for this part. In general, it is shown that a ring \(R\) whose idempotents are central, is \(\pi\)-regular if and only if for some two-sided nil ideal \(I\) of \(R\), \(R/I\) is \(\pi\)-regular. As a particular case, this holds when \(I\) is the prime radical of \(R\).
Several other related results are also included, such as: Let \(R\) be an abelian \(\pi\)-regular ring and \(P\) be a prime ideal of \(R\). Then every element of \(R/P\) is either nilpotent or a unit. In particular, if \(P\) is a prime ideal containing \(\text{Nil}(R)\), then \(R/P\) is a division ring.

MSC:

16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
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