A note on primeness in near-rings and matrix near-rings (Q1917700)

Matrix near-rings were introduced by \textit{J. D. P. Meldrum} and \textit{A. P. J. van der Walt} [Arch. Math. 47, 312-319 (1986; Zbl 0611.16025)]. Let \(N\) be a right near-ring with unity. Define \(f^r_{ij}:N^n\to N^n\) (\(1\leq i,j\leq n,\;r\in N\)) by \(f^r_{ij}(x_1,\dots,x_n)=(y_1,\dots,y_n)\), where \(y_k=0\) if \(k\neq i\) and \(y_k=rx_j\) if \(k=i\). The matrix near-ring \(M_n(N)\) is the subnear-ring of \({\mathcal M}(N^n,N^n)\) generated by all the \(f^r_{ij}\)'s. If \(I\subseteq N\), \(I^*=\{A\in{\mathcal M}_n(N)\mid A\rho\in I^n \forall\rho\in N^n\}\). A number of new concepts are introduced in this paper. A left ideal \(P\) of \(N\) is called prime if \(A\), \(B\) left ideals of \(N\), \(AB\subseteq P\) implies \(A\subseteq P\) or \(B\subseteq P\). \(N\) is called strictly equiprime if for each \(0\neq a\in N\), there exists \(x_a\in N\) such that \(ax_a u=ax_av\) implies \(u=v\), for all \(u,v\in N\). If \(A,B\in{\mathcal M}_n(N)\), then \(A\) and \(B\) are called \(j\)-th row equivalent if \((f^1_{1j}+\dots+f^1_{nj})A=(f^1_{1j}+\dots+f^1_{nj})B\). \({\mathcal M}_n(N)\) is said to be \(j\)-th row strictly equiprime if for each \(0\neq A\in{\mathcal M}_n(N)\), there exists \(\phi\in{\mathcal M}_n(N)\) such that \(A\phi U=A\phi V\) implies \(U\) and \(V\) are \(j\)-th row equivalent, for all \(U,V\in{\mathcal M}_n(N)\). A number of natural characterizations of prime left ideals are given, and the main results of the paper are as follows: Proposition 3.1. If \(N\) is a strictly equiprime near-ring with unity, then \({\mathcal M}_n(N)\) is \(j\)-th row strictly equiprime for \(1\leq j\leq n\). Proposition 3.3. If \(I\) is a prime left ideal of \(N\), then \(I^*\) is a prime left ideal of \({\mathcal M}_n(N)\).