Maximally-fine rings (Q6087858)

In an associative ring \(R\) with unity, an element \(a \in R\) is \textit{fine} when it is the sum of a unit and a nilpotent from the ring \(R\) and \(R\) is a \textit{fine} ring if it is a nonzero ring where all its nonzero elements are fine. Further if the equality \(RaR = R\) holds, \(a\) is said to be a \textit{full} element of \(R\). A ring \(R\) is \textit{maximally-fine} (or \textit{\(m\)-fine} for short) when for each \(a \in R\), if \(a\) is full, then \(a\) is fine. In this article the authors have investigated rings where every full element is fine. In particular, it was shown that the endomorphism ring of any vector space has this property, so such rings are as fine as possible. Consequently, they were able to construct new examples of fine rings.