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.

1-Primitive Near-rings

“…We will see in this paper that this restriction is not needed. Thus we can generalize the result of [4] to near-rings not necessarily having a multiplicative right identity and obtain a description of all zero symmetric 1-primitive near-rings as well as 2-primitive near-rings as dense subnear-rings of sandwich centralizer near-rings. This construction simplifies the construction of [5].…”

“…where X is a non-empty set, (N, +) a group, φ the sandwich function, B a subgroup of Aut(N, +), S a group of permutations on X and ψ ∈ Hom(S, B). This construction is more technical than that in [4] and that we will use in our approach in this paper and the functions in M(X , N, φ, ψ, B,C) are not centralized by elements of S. For the details of the construction we refer the interested reader to [5]. The idea of combining the concepts of centralizer near-rings and sandwich nearrings used in this paper allows us to explicitely describe and construct the sandwich function φ which determines the multiplication in the primitive near-ring.…”

“…If one studies primitive near-rings without necessarily having an identity, then still results are available but much more technical in detail in comparison to the 2-primitive case with identity. Combination of the concepts of centralizer near-rings and sandwich near-rings were used in [4] and in [5] to describe 1-primitive near-rings which do not necessarily have an identity. In this paper we will extend and simplify the results obtained in [4] and [5] and we will also consider 2-primitive near-rings as a special case of 1-primitive near-rings.…”

“…Combination of the concepts of centralizer near-rings and sandwich near-rings were used in [4] and in [5] to describe 1-primitive near-rings which do not necessarily have an identity. In this paper we will extend and simplify the results obtained in [4] and [5] and we will also consider 2-primitive near-rings as a special case of 1-primitive near-rings. We will now briefly give the definitions for zero symmetric primitive near-rings to settle our notation which we will keep throughout the paper.…”

“…As in the example before the function h : N → M 0 (X , Z 4 , φ), f → ψ f is easily seen to be a near-ring isomorphism. Combinations of the concepts of centralizer near-rings and sandwich near-rings were used in [4] and in [5] to describe 1-primitive near-rings which do not necessarily have an identity. In [4], 1-primitive near-rings are described as dense subnear-rings of near-rings of the type of M 0 (X , Γ, φ, S) := { f : X −→ Γ | f (0) = 0 and ∀s ∈ S ∀x ∈ X : f (s(x)) = s( f (x))}, see Definition 2.1, where S is an automorphism group of Γ acting without fixed points on the set X \ {0} and the near-ring operations are pointwise + of functions and sandwich multiplication, but the result and proof in [4] requires that the primitive near-ring has a multiplicative right identity.…”