Partial Order in Matrix Nearrings

Let N be a zero-symmetric (right) nearring with identity. We introduce a partial order in the matrix nearring corresponding to the partial order (defined by Pilz [10]) in N. A positive cone in a matrix nearring is defined and a characterization theorem is obtained. For a convex ideal I in N, we prove that the corresponding ideal I∗ is convex in Mn(N), and conversely, if I is convex in Mn(N), then I∗ is convex in N. Consequently, we establish an order-preserving isomorphism between the p.o. quotient matrix nearrings Mn(N)/I∗ and Mn(N0)/(I0)∗ where I and I0 are the convex ideals of p.o. nearrings N and N0 respectively. Finally, we prove some properties of Archimedian ordering in matrix nearrings corresponding to those in nearrings.