Matrix near rings
Until this article, there has not been an acceptable approach to the concept of a near-ring of matrices over an arbitrary near-ring. The authors overcome the inherent problems associated with arrays and are motivated by the fact that for a ring, each matrix represents an endomorphism of \((R^ n,+)\) and as such it is derived from the endomorphisms of the module \(RR\). To this end, the near-ring \(M_ n(R)\) of \(n\times n\) matrices over a near-ring \(R\) with unity is defined as the subnear-ring of all mappings of \(R^ n\to R^ n\) generated by the set \(\{f^ r_{ij}\mid r\in R\), \(1\leq i,j\leq n\}\) such that \(f^ r_{ij}=\iota_ if^ r\pi_ j\) for \(\pi_ j\) the projection and \(\iota_ i\) an injection on the elements of \(R^ n\) and \(f^ r(s)=rs\) for all \(s\in R\). Then \(M_ n(R)\) is a right near-ring with unity. If \(R\) is a ring with unity, then \(M_ n(R)\) is isomorphic to the ring of matrices over \(R\). If \(R\) is a near-ring without unity, compensation is made as follows: Adjoin \(\alpha\) to \((R,+)\) to obtain \((R_{\alpha},+)\). Then the set of all functions of \(R_{\alpha}\) into \(R\) is a near-ring \(M(R_{\alpha})\) with respect to pointwise addition and composition. Then embed \(R\) into \(M(R_{\alpha})\) by \(r\to f^ r\) such that \(f^ r(s)=rs\) if \(s\in R\) and \(f^ r(\alpha)=r\). Proceed as before. Some fundamental rules for matrix calculations are developed from which follow various conclusions such as these: If \(R\) is zerosymmetric, or d.g., or an abstract near-ring, then so is \(M_ n(R)\). A portion of this investigation is given to the relationship of two-sided ideals of \(R\) with those of \(M_ n(R)\). This is carefully developed. Among the results we find that \(R\) is simple if and only if \(M_ n(R)\) is simple and that if \(R\) is a prime (semiprime) near-ring, so is \(M_ n(R)\). The presentation is clear and the article is self-contained.
scientific article; zbMATH DE number 3987510