Matrix near-rings and 0-primitivity

Abstract

In this article, we study the implication of the primitivity of a matrix near-ring ${\mathbb{M}_n(R) (n >1 )}$ and that of the underlying base near-ring R. We show that when R is a zero-symmetric near-ring with identity and ${\mathbb{M}_n(R)}$ has the descending chain condition on ${\mathbb{M}_n(R)}$-subgroups, then the 0-primitivity of ${\mathbb{M}_n(R)}$ implies the 0-primitivity of R. It is not known if this is true when the descending chain condition on ${\mathbb{M}_n(R)}$ is removed. On the other hand, an example is given to show that this is not true in the case of generalized matrix near-rings.