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Near-rings generated by \(R\)-modules. (English) Zbl 0811.16036
Extending 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.
16Y30 Near-rings
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16N80 General radicals and associative rings
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