On strongly \(\pi\)-regular ideals.
For an associative ring \(R\) with identity, the authors introduce the notion of strongly \(\pi\)-regular ideal as an ideal \(I\) of \(R\) with the property that for any \(x\in I\), there exist a natural number \(n\) and an element \(y\in I\) such that \(x^n=x^{n+1}y\). They show that every regular square matrix over a strongly \(\pi\)-regular ideal of a ring is unit-regular. This result allows them to prove that every regular square matrix over a strongly \(\pi\)-regular ideal of a ring admits a diagonal reduction. Also, for a strongly \(\pi\)-regular ideal \(I\) of \(R\) and a natural number \(n\), it is shown that \(R\) is isomorphic to some \(n\times n\) matrix ring if and only if \(R/I\) is isomorphic to some \(n\times n\) matrix ring.
scientific article; zbMATH DE number 2137305
scientific article