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In this paper, we consider an R-group where R is a zero-symmetric right nearring. We define generalized essential ideal of an R-group and prove several properties. Further, we extend this notion to obtain a one-one correspondence between s-essential ideals of R-group and those of Mn(R)-group Rn. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an equiprime fuzzy graph of a nearring with respect to the level ideal of a fuzzy ideal. We interrelate graph theoretical properties of the graph and ideal theoretical properties of nearring. We show that the properties like vertex cut, connectedness of the graph depend on the properties of the fuzzy ideal. We define ideal symmetry of the graph and find conditions for the graph to be ideal symmetric. If the fuzzy ideal is equiprime then we show that the level set induces a fuzzy clique. We find conditions required for the level set to be the vertex cover of the graph. We find interrelation between equiprime fuzzy graph and fuzzy graph of nearring with respect to level ideal. We study properties of the graph under neaaring homomorphism. We prove that the connectedness of the graph in homomorphic image depends on properties of ideal. We obtain conditions required for homomorphic image of an equiprime fuzzy ideal to be an equiprime fuzzy ideal. [ABSTRACT FROM AUTHOR]

We consider an R-group G, where R is a zero symmetric right nearring. We obtain the Ω-dimension of sum of two ideals of G, as a natural generalization of sum of two subspaces of a finite dimensional vector space; indeed, difficulty due to non-linearity in G. However, in this paper we overcome the situation under a suitable assumption. More precisely, we prove that for a proper ideal of G with -finite Goldie dimension (- FGD), if K1,K2 are ideals of G wherein K1 ∩K2 is an Ω-complement, then dimΩ(K1 + K2) = dimΩ(K1) + dim (K2) - dim (K1 ∩ K2). In the sequel, we prove several properties. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of ΘΓ N-group as a generalization of algebraic structures of N-group and gamma nearring. We present motivating examples of ΘΓ N-groups and prove classical isomorphism theorems. [ABSTRACT FROM AUTHOR]