Superfluous ideals of N-groups

Abstract

AbstractWe consider a right nearring N and a module over N (known as, N-group). For an arbitrary ideal (or N-subgroup) $$\varOmega $$ Ω of an N-group G, we define the notions $$\varOmega $$ Ω -superfluous, strictly $$\varOmega $$ Ω -superfluous, g-superfluous ideals of G. We give suitable examples to distinguish between these classes and the existing classes studied in Bhavanari (Proc Japan Acad 61-A:23–25, 1985; Indian J Pure Appl Math 22:633–636, 1991; J Austral Math Soc 57:170–178, 1994), and prove some properties. For a zero-symmetric nearring with 1, we consider a module over a matrix nearring and obtain one-one correspondence between the superfluous ideals of an 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 N. Furthermore, we define a graph of superfluous ideals of a nearring and prove some properties with necessary examples.