Strongly Prime Near-Ring Modules

Abstract

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).

Generalization of prime ideals in $$M_n(N)$$-group $$N^{n}$$

“…Juglal et.al [7] studied different prime N-ideals and prime relations between generalized matrix nearring and multiplication modules over a nearring. Further, Juglal and Groenewald [8] studied the class of strongly prime nearring modules and shown that it forms a -special class. In Bhavanari and Kuncham [9], the relation between the ideals of the N-group N and the ideals of M n (N)-group N n has been studied.…”

2-Absorbing R-ideals of modules over near rings

“…However, even a near ring is a zero symmetric near ring, an R-subgroup may not be an ideal. Then K is a zero symmetric near ring, {e} is the only one ideal of K and {e, b} is an R-subgroup of K, see [9]. Therefore, {e, b} is an R-subgroup of K but is not an ideal of K even K is a zero symmetric near ring.…”