Near-rings generated by \(R\)-modules.

*(English)*Zbl 0811.16036Extending an idea used by 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.

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.

Reviewer: J.D.P.Meldrum (Edinburgh)

##### MSC:

16Y30 | Near-rings |

16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |

16N80 | General radicals and associative rings |