Superfluous ideals of N-groups

Abstract

We consider a right nearring N and a module over N (known as, N-group). For an arbitrary ideal (or N-subgroup) varOmega of an N-group G, we define the notions varOmega -superfluous, strictly varOmega -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)-group N^{n}, where 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.