On Jordan ideals and semigroup ideals in 3-prime near-rings with multiplicative derivations

In the current paper, we study the structure of Jordan ideals of a 3-prime near-ring which satisfies some algebraic identities involving left generalized derivations and right centralizers. The limitations imposed in the hypothesis were justified by examples.

The purpose of this paper is to study derivations and generalized derivations satisfying certain identities on semigroup ideals of near-rings. Some well-known results characterizing commutativity of 3-prime near-rings by derivations have been generalized by using semigroup ideals. Moreover, examples proving the necessity of the 3-primeness hypothesis are given.

Let $$\mathcal {N}$$ be a 3-prime near-ring with center $$Z(\mathcal {N})$$ and $$\mathcal {J}$$ a nonzero Jordan ideal of $$\mathcal {N}$$ . The aim of this paper is to prove some theorems showing that $$\mathcal {N}$$ must be commutative if it admits a left multiplier F satisfying any one of the following properties: $$(i)\,F(\mathcal {J})\subseteq Z(\mathcal {N})$$ , $$(ii)\,F(\mathcal {J}^{2})\subseteq Z(\mathcal {N})$$ , $$(iii)\,F(ij)+[i, j]\in Z(\mathcal {N})$$ , $$(vi)\,F(ij)-ij+ji\in Z(\mathcal {N})$$ , $$(v)\,F(i\circ j)\in Z(\mathcal {N})$$ and $$(vi)\,F(i)G(j)\in Z(\mathcal {N}),$$ for all $$i, j\in \mathcal {J}.$$ Moreover, we give some examples which show that the hypotheses placed in our results are not superfluous.

We will start this article by proving a crucial concept, which will allow us to overcome a set of obstacles we encountered in previous articles concerning the commutativity of near-ring involving homoderivations and Jordan ideals. Furthermore, we present examples to show that limitations imposed in the hypothesis of our results are necessary.

Jordan ideals and $$(\alpha , \beta )$$-derivations on 3-prime near-rings and rings

We will extend in this paper some results about commutativity of Jordan ideals proved in [2] and [6]. However, we will consider left derivations instead of derivations, which is enough to get good results in relation to the structure of near-rings. We will also show that the conditions imposed in the paper cannot be removed.

The purpose of this paper is to study derivations satisfying certain differential identities on Jordan ideals of 3-prime near-rings. Moreover, we provide examples to show that hypothesis of our results are necessary.

In this paper, we investigate commutativity of rings with involution in which derivations satisfy certain algebraic identities on Jordan ideals. Moreover, we extend some results for derivations of prime rings to Jordan ideals. Furthermore, an example is given to prove that the ∗-primeness hypothesis is not superfluous.

In the present paper, we investigate the commutativity of 3-prime near-rings satisfying certain conditions involving left generalized multiplicative derivations on semigroup ideals. Moreover, examples have been provided to justify the necessity of 3-primeness condition in the hypotheses of various results.

In the present paper we investigate commutativity in prime rings and 3-prime near-rings admitting a generalized derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings and 3-prime near-rings have been generalized.

In the present paper, we investigate the notion of generalized derivation satisfying certain algebraic identities in 3-prime near-ring N which forces N to be a commutative ring. Moreover, an example proving the necessity of the primeness of N is given.

In the present paper, we investigate the commutativity of addition and ring behavior of $3$-prime near-rings satisfying certain conditions involving generalized $n$-derivations on semigroup ideals. Moreover, examples justifying the necessity of the $3$-primeness condition in all the results are provided.

The current paper studied the concept of right n-derivation satisfying certified conditions on semigroup ideals of near-rings and some related properties. Interesting results have been reached, the most prominent of which are the following: Let M be a 3-prime left near-ring and A_1,A_2,…,A_n are nonzero semigroup ideals of M, if d is a right n-derivation of M satisfies on of the following conditions,d(u_1,u_2,…,(u_j,v_j ),…,u_n )=0 ∀ 〖 u〗_1 〖ϵA〗_1 ,u_2 〖ϵA〗_2,…,u_j,v_j ϵ A_j,…,〖u_n ϵA〗_u;d((u_1,v_1 ),(u_2,v_2 ),…,(u_j,v_j ),…,(u_n,v_n ))=0 ∀u_1,v_1 〖ϵA〗_1,u_2,v_2 〖ϵA〗_2,…,u_j,v_j ϵ A_j,…,〖u_n,v_n ϵA〗_u ;d((u_1,v_1 ),(u_2,v_2 ),…,(u_j,v_j ),…,(u_n,v_n ))=(u_j,v_j ) ∀ u_1,v_1 〖ϵA〗_1 ,u_2,v_2 〖ϵA〗_2,…,u_j,v_j ϵ A_j,…,〖u_n,v_n ϵA〗_u;If d+d is an n -additive mapping from A_1×A_2×…×A_n to M;d(u_1,u_2,…,(u_j,v_j ),…,u_n)∈ Z(M) ∀〖 u〗_1 〖ϵA〗_1 ,u_2 〖ϵA〗_2,…,u_j,v_j ϵ A_j,…,〖u_n ϵA〗_u,; d((u_1,v_1 ),(u_2,v_2 ),…,(u_j,v_j ),…,(u_n,v_n ))∈ Z(M) ∀ u_1,v_1 〖ϵA〗_1 ,u_2,v_2 〖ϵA〗_2,…,u_j,v_j ϵ A_j,…,〖u_n,v_n ϵA〗_u;Then M is a commutative ring.

In this paper, our main objective is to introduce the notion of a fuzzy semigroup ideal by using Yuan and Lee's definition of fuzzy group based on fuzzy binary operations. Also, some of its basic properties are studied analogously to the known results in the case of semigroup ideals defined in the framework of ordinary near-rings.

This paper studies homoderivations satisfying certain conditions on semigroup ideals of near-rings. In addition, we include some examples of the necessity of the hypotheses used in our results.