Structure of periodic near rings (Q2746292)

The theorem of this paper deals with distributively-generated near-rings \(N\) satisfying one of seven conditions of the form \(xy=q(x,y)\), where \(q(x,y)\) denotes an element of the subnear-ring generated by \(x\) and \(y\) -- for example, condition \((\text{P}_2)\): for each \(x,y\in N\), there exists \(n=n(x,y)\geq 1\) such that \(xy=(yx)^np(yx)\), where \(p(yx)\) is some sum of powers \((yx)^k\), \(k\geq 2\), and additive inverses of such powers. In every case, \(N\) is periodic and commutative and can be written as an orthogonal sum of a \(J\)-ring and a near-ring with trivial multiplication. These results extend results of the reviewer and \textit{S. Ligh} [Math. J. Okayama Univ. 31, 93-99 (1989; Zbl 0696.16029)].