A study on near-rings with \(SR\)-conditions (Q2755636)

The \(SR\)-conditions of the title for a near-ring \(N\) are (i) \(a,b\in N\) and \(ab=0\) implies \(ba=b0\); (ii) \(a\in N\) and \(a^3=a^2\) implies \(a^2=a\). This paper has two main results. The first shows that for a zero symmetric near-ring \(N\) the following are equivalent (a) \(N\) is left \(s\)-unital (\(a\in Na\;\forall a\in N\)) and left bipotent (\(Na=Na^2\)); (b) \(N\) is reduced and left bipotent; (c) \(N\) is left strongly regular (\(\forall a\in N\;\exists x\in N\) such that \(a=xa^2\)); (d) \(N\) is regular and \(\forall a\in N\;\exists x\in N\) such that \(ax=xa\). The second shows that if \(N\) satisfies the \(SR\)-conditions, then \(N\) is \(\pi\)-regular if and only if \(N\) is strongly \(\pi\)-regular. These near-rings are also linked to a generalized version of left bipotent.