On the structure of some rings and nearrings (Q2777546)

It is proved that a ring \(R\) is a direct sum of a \(J\)-ring and a zero ring if for each \(x,y\in R\) there exist nonnegative integers \(p\), \(q\) and \(f(t)\in t^2\mathbb{Z}[t]\) such that (*) \(xy=x^pf(xy^2x)x^q\); and there are some extensions in which (*) is merely assumed to hold for \(x\), \(y\) in some proper subset of \(R\). There is also an orthogonal sum decomposition theorem for near-rings \(N\) such that for each \(x,y\in N\) there exist nonnegative integers \(p\), \(q\), \(r\) with \(r>1\) for which \(xy=x^p(xy^2x)^rx^q\). These results are motivated by theorems of the reviewer and \textit{S. Ligh} [Math. J. Okayama Univ. 31, 93-99 (1989; Zbl 0696.16029)].