Endomorphisms and distributive elements in near-rings determined by rings and modules
Let \(R\) be a ring with identity, \(V\) a faithful, unital \(R\)-module and let \(N=M_R(V)\) be the centralizer nearring determined by \((R,V)\). Denote by \(\text{Dist}(N)\) the set of distributive elements of \(N\). The author is interested in conditions which force \(\text{End}_R(V)=\text{Dist}(N)\) as well as conditions which force \(N=M_R(V)=\text{Dist}(N)\). An \(R\)-module \(V\) is called \(N\)-cyclic (locally \(N\)-cyclic) if \(\exists v\in V\) such that \(Nv=V\) (if \(\forall v,w\in V\) \(\exists z\in V\) such that \(v,w\in Nz\)). If \(N\) is locally \(N\)-cyclic (or \(N\)-cyclic) then \(\text{Dist}(N)=\text{End}_R(V)\). It is well known [see \textit{J. Hausen} and \textit{J. A. Johnson}, J. Aust. Math. Soc. 59, No. 2, 173-183 (1995; Zbl 0852.16032)] that if \(V\) is locally cyclic (or cyclic) as \(R\)-module then \(N=\text{End}_R(V)\). In this paper it is shown that if \(V\) is a finite direct sum of locally cyclic \(R\)-modules then \(\text{End}_R(V)=\text{Dist}(N)\). Moreover if \(R\) is an integral domain then \(N\) is a ring iff either \(V\) is cyclic or \(V\) is torsion as \(R\)-module. If \(R\) is a Dedekind domain and \(V\) is a finitely generated \(R\)-module then it is well known that \(V\simeq R/{I_1}\oplus R/{I_2}\oplus\cdots\oplus R/{I_n}\oplus I\oplus R^m\) where all \(I_j\) and \(I\) are ideals of \(R\). So in these cases one has \(\text{Dist}(N)=N\) if and only if \(V\) is torsion, cyclic or isomorphic to some ideal of \(R\).
scientific article; zbMATH DE number 1563150
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