Let N be a 3-prime near-ring with the center Z(N), and U be a nonzero semigroup ideal of N. In the present paper it is shown that a 3-prime near-ring N is a commutative ring if and only if it admits left multipliers F and G satisfying any one of the following properties: (i)F(x)G(y)±[x, y] ∈ Z(N); (ii)F(x)G(y)±x•y ∈ Z(N); (iii) F(x)G(y) ± yx ∈ Z(N); (iv) F(x)G(y) ± xy ∈ Z(N) and (v) F([x, y]) ± G(x • y) ∈ Z(N) for all x, y ∈ U .

Let P be a poset and d be a derivation on P . In this research, the notion of generalized d-derivation on partially ordered sets is presented and studied. Several characterization theorems on generalized d-derivations are introduced. The properties of the fixed points based on the generalized d-derivations are examined. The properties of ideals and operations related with generalized d-derivations are studied.2010 MSC: 06E20, 13N15, 06Axx.

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In this paper, as a generalization of derivation on a partially ordered set, the notion of a triple derivation is presented and studied on a partially ordered set. We study some fundamental properties of the triple derivation on partially ordered sets. Moreover, some examples of triple derivations on a partially ordered set are given. Furthermore, it is shown that the image of an ideal under triple derivation is an ideal under some conditions. Also, the set of fixed points under triple derivation is an ideal under certain conditions. We establish a series of further results of the following nature. Let ( , ) be a partially ordered set. 1. If , are triple derivations on , then = if and only if Fix ( ) = Fix ( ). 2. If is a triple derivation on , then, for all ∈ ;Fix ( ) ∩ ( ) = ( ( )).3. If and are two triple derivations on , then and commute. 4. If and are two triple derivations on , then if and only if = . In the end, the properties of ideals and operations related to triple derivations are examined.