On unit-regular ideals. (Q1419677)

The authors introduce the notion of unit-regular ideal in a unital ring. This generalizes the notion of a unit-regular ring. They give a necessary and sufficient condition for a regular ideal \(I\) of a unital ring \(R\) to be unit-regular in terms of elements of \(I\). They prove that every square matrix over a unit-regular ideal of a unital ring is a product of an idempotent matrix and an invertible matrix. They also prove that every square matrix over a unit-regular ideal of a unital ring admits a diagonal reduction. Finally, they prove that a regular ideal \(I\) of a unital ring \(R\) is unit-regular if and only if the pseudo-similarity via \(I\) implies similarity.