Primitive Near-rings — Some Structure Theorems

Abstract

We show that any zero symmetric 1-primitive near-ring with descending chain condition on left ideals can be described as a centralizer near-ring in which the multiplication is not the function composition but sandwich multiplication. This result follows from a more general structure theorem on 1-primitive near-rings with multiplicative right identity, not necessarily having a chain condition on left ideals. We then use our results to investigate more closely the multiplicative semigroup of a 1-primitive near-ring. In particular, we show that the set of regular elements forms a right ideal of the multiplicative semigroup of the near-ring.