It is well known that there are several non-equivalent types of prime near-rings which are all equivalent in the case of associative rings. In this paper we introduce various characterizations of prime modules in a zero-symmetric near-ring R. The connection of a prime R-ideal P of a module M and the ideal (P:M) of the near-ring R is also investigated.

The notion of a matrix nearring over an arbitrary nearring was introduced by (Meldrum and Walt Arch. Math. 47(4): 312–319, 1986). In this paper, we define the notions such as weakly $$\tau$$ τ -prime $$(\tau =0,c,3,e)$$ ( τ = 0 , c , 3 , e ) ideals of an N-group G,  which are the generalization of the classes of $$\tau$$ τ -prime ideals of G, and provide suitable examples to distinguish between the two classes. We extend the concept to obtain the one-one correspondence between weakly $$\tau$$ τ -prime ideals $$(\tau =0,c,3,e)$$ ( τ = 0 , c , 3 , e ) of N-group (over itself) and those of $$M_n(N)$$ M n ( N ) -group $$N^{n}$$ N n , where $$M_n(N)$$ M n ( N ) is the matrix nearring over the nearring N. Further, we prove the correspondence between weakly 2-absorbing ideals of these classes.

The flow (or lack thereof) of several kinds of primeness between a zerosymmetric near-ring R and its group near-ring R[G] for certain groups G is discussed. In certain cases, results are contrasted against what happens in the matrix near-ring situation.2000 Mathematics Subject Classification: 16Y30

We introduce the notion of a strongly prime near-ring module and then characterize strongly prime near-rings in terms of strongly prime modules. Furthermore, we define a T -special class of near-ring modules and then show that the class of strongly prime modules forms a T -special class. T -special classes of strongly prime modules are then used to describe the strongly prime radical. Basic ConceptsA (right) near-ring is a triple (N , +, ·) such that (N , +) is a group (not necessarily abelian), (N , ·) is a semigroup and the (right) distributive law (x + y)z = xz + yz holds.Let R be a near-ring and (M, +) be a group. Then M is called an R-module if for all r 1 , r 2 ∈ R and m ∈ M, it follows that (r 1 + r 2 )m = r 1 m + r 2 m and (r 1 r 2 )m = r 1 (r 2 m). If R is a near-ring, then an obvious R-module is the group (R, +). We shall denote this particular R-module by R R.Let H be a subgroup of M such that for all r ∈ R and h ∈ H , we have that rh ∈ H . Then H is called an R-submodule of M. (We shall denote this by H ≤ R M). An R-ideal of M is a normal subgroup P of M such that for all r ∈ R, m ∈ M and n ∈ P, r (m + n) − rm ∈ P. (We shall denote this as P R M).

Le Riche, Meldrum and Van der Walt introduced the concept of a group near-ring. Thereafter, starting with an ideal in the base near-ring, they constructed two corresponding ideals in the group near-ring. Furthermore, by starting with an ideal in the group near-ring, they constructed a corresponding ideal in the base near-ring. In this we paper, we generalise both the concept of the group near-ring and the construction of the three ideals. We introduce a fourth ideal, and we investigate some relationships amongst these four ideals. We also investigate whether a prime condition placed on an R-ideal of a near-ring module implies that prime condition on the corresponding ideal in the generalised group near-ring and vice versa.