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On a decomposition of near-rings in a subdirect sum of near-fields. (English) Zbl 0638.16030

A proper right ideal B of a left zero symmetric near-ring \((R,+,\cdot)\) is called strictly maximal if it is maximal as an R-subgroup, i.e., B is not properly contained in any proper subgroup K of \((R,+)\) such that kx\(\in K\) for all \(k\in K\), \(x\in R\). The near-ring \((R,+,\cdot)\) is said to have property (D) if whenever P is a strictly maximal right ideal of \((R,+,\cdot)\) we have qR\(\not\subseteq P\) for all \(q\not\in P\). The author studies such near-rings. He gives three equivalent statements on a strictly maximal right ideal of such a near-ring. He also derives necessary and sufficient conditions for a near-ring in which every non- zero ideal, as a near-ring, has property (D) to be isomorphic to a subdirect sum of near-fields.
Misprint: Change ‘P\(=Q'\) to ‘R\(=Q'\) on line 8 of page 44.
Reviewer: S.Baskaran

MSC:

16Y30 Near-rings
12K05 Near-fields
16Dxx Modules, bimodules and ideals in associative algebras
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