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Nearring multiplications on groups. (English) Zbl 0832.16036

This paper is concerned with finding additive groups which only support trivial nearring multiplications. This is a problem which has attracted a lot of attention over a long period and is still unsolved, although this paper has considerably increased our knowledge of this area.
The main part of the paper uses a semigroup theoretical approach and is mainly concerned with \(S\)-acts. Here we are considering an additive group \(G\), and a semigroup \(S\) acting as endomorphisms of \(G\). A number of results which seem at first sight to be rather technical, are established. From these flow a large number of results showing that many groups support non-trivial nearring multiplications. This enables the authors to restrict quite substantially the structure of a group which has only trivial nearring multiplications.
The last two topics differ somewhat from the main part. First a model theoretic approach to the problem shows that any group that has only trivial nearring multiplications cannot be described by first order properties. Then an application to centralizer nearrings characterises a class of such nearrings which have no non-trivial idempotents.

MSC:

16Y30 Near-rings
16N80 General radicals and associative rings
20M20 Semigroups of transformations, relations, partitions, etc.
20M50 Connections of semigroups with homological algebra and category theory
16B70 Applications of logic in associative algebras
20E36 Automorphisms of infinite groups
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