an:00528843
Zbl 0811.16036
Clay, James R.; Kautschitsch, Hermann
Near-rings generated by \(R\)-modules
EN
Math. Pannonica 4, No. 2, 287-297 (1993).
00017865
1993
j
16Y30 16W60 16N80
formal power series; generalized near-rings; ideals; \(J\)-radicals; prime and nil radicals
Extending an idea used by \textit{H. Gonshor} [Pac. J. Math. 14, 1237-1240 (1964; Zbl 0128.025)], the authors construct from a ring \(R\) and an \(R\)- module \(M\) near-rings whose underlying set is \(M \times R\). They start with formal power series \(R[[x]]\) over a ring \(R\). By considering those with zero constant term and factoring out the principal ideal generated by \(x^ k\) for \(k = 1,2,3\), an ideal which is both a ring ideal and a near-ring ideal in the near-ring defined on power series when composition is the composition of maps, various near-rings are obtained. The technique can be generalized and a good deal of information about the structure of these generalized near-rings is obtained.
The authors then consider in more detail the case which generalizes the situation when \(k = 1\), and which is mentioned at the beginning of this review. They analyse in detail the ideals of such a near-ring, obtaining a substantial amount of detailed information. This is used to identify the four \(J\)-radicals of the near-ring as well as the prime and nil radicals. The development is interesting and could well lead to further work in this area.
J.D.P.Meldrum (Edinburgh)
Zbl 0128.025