The \(J_ 0\)-radical of matrix near-rings (Q804681)

The main result is: Let \((R,+,.)\) be a weakly distributive d.g. right near-ring with identity. Then \(J_ 0(R)^*=J_ 0(M_ n(R))\). Notation: if \(R^ n\) is the direct sum of n copies of \((R,+)\), let \(M(R^ n)\) be the near-ring of all mappings of the group \(R^ n\) into itself (under pointwise addition and composition of mappings); \(M_ n(R)\) is then defined to be the subnear-ring of \(M(R^ n)\) generated by the set of all mappings \(\iota_ i\lambda (r)\pi_ j\), where \(r\in R\), \(1\leq i,j\leq n\), \(\lambda\) (r): \(R\to R\) is left multiplication by r, \(\pi_ j\) is the j-th coordinate projection \(R^ n\to R\), and \(\iota_ i\) is the i-th coordinate injection \(R\to R^ n\); for a near-ring R, the authors define \(J_ 0(R)\) to be the intersection of all annihilators of monogenic simple (left) R-modules; for an ideal I of R, they define \(I^*=\{A\in M_ n(R)\); \(AR^ n\subseteq I^ n\}\).