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 in Near-rings: the theory and its applications, North Holland, Amsterdam, 1983) 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 M n ( N ) , and conversely, if I is convex in M n ( N ) , then I ∗ is convex in N . Consequently, we establish an order-preserving isomorphism between the p.o. quotient matrix nearrings M n ( N ) / I ∗ and M n ( N ′ ) / ( I ′ ) ∗ where I and I ′ are the convex ideals of p.o. nearrings N and N ′ , respectively. Finally, we prove some properties of Archimedean ordering in matrix nearrings corresponding to those in nearrings.