On group near-rings
In this paper the authors define group near-rings for the general case and lay the foundations for further development of the subject. Let \(R\) be a near-ring with identity 1 and \(G\) a group with identity \(e\). \(R^ G\) denotes the cartesian direct sum of \(|G|\) copies of \((R,+)\) indexed by the elements of \(G\). \(M(R^ G)\) is the right near-ring of all mappings of the group \(R^ G\) into itself. For \(r\in R\) and \(g\in G\) by \([r,g]\) we denote the mapping in \(M(R^ G)\) defined by \(([r,g](\mu))(h)=r_{\mu}(hg)\) for all \(\mu\in R^ G\), \(h\in G\). The set \(\{[r,g]\mid r\in R\), \(g\in G\}\) generates a subnear-ring of \(M(R^ G)\) which will be denoted by \(R[G]\) and called the group near-ring constructed from \(R\) and \(G\). If \(R\) happens to be a ring, then \(R[G]\) is isomorphic to the standard group-ring constructed from \(R\) and \(G\). If \(R\) is distributively generated, then so is \(R[G]\) for every group \(G\). Let \(\phi: R\to S\) be an epimorphism of near-rings and \({\bar \phi}:R^ G\to S^ G\) be the epimorphism defined by \({\bar\phi}(\mu)(g)= \phi(\mu(g))\) for all \(g\in G\). We can define an epimorphism \(\phi^*: R[G]\to S[G]\) by \(\phi^*(A)(\sigma)={\bar \phi}(A{\tilde \phi}(\sigma))\) for all \(\sigma \in S^ G\), where \({\tilde \phi}\) is a right inverse of \({\bar \phi}\). Put \((\text{Ker\,}\phi)^*=\text{Ker\,}\phi^*\). The authors show that the mapping \((\;)^*\) is an injection from the set of ideals of \(R\) into that of \(R[G]\). Moreover, \(({\mathcal A}\cap {\mathcal B})^*={\mathcal A}^*\cap {\mathcal B}^*\) and \({\mathcal A}^*+{\mathcal B}^*\subseteq ({\mathcal A}+{\mathcal B})^*\). Let \({\mathcal A}^*\) be the ideal generated by the set \(\{[a,e]\mid a\in {\mathcal A}\}\). Then the mapping \((\;)^+\) is an injection from the set of ideals of \(R\) into that of \(R[G]\). Moreover, \({\mathcal A}^+\subseteq {\mathcal A}^*\), \(({\mathcal A}+{\mathcal B})^+={\mathcal A}^++{\mathcal B}^+\) and \(({\mathcal A}\cap {\mathcal B})^+\subseteq {\mathcal A}^++{\mathcal B}^+\). Denote by \(\eta\) the function in \(R^ G\) such that \(\eta (e)=1\) and \(\eta (g)=0\) if \(g\neq e\). For an ideal \({\mathcal A}\) in \(R[G]\), put \({\mathcal A}_*=\{a\in R\mid a=(A\eta)(e)\) for some \(A\in {\mathcal A}\}\). Then \({\mathcal A}\subseteq ({\mathcal A}_*)^*\).
scientific article; zbMATH DE number 4097583