In the current study, the definition of mapping of fuzzy neutrosophic generalized semi-continuous and fuzzy neutrosophic alpha has generalized mapping as continuous. The study confirmed some theorems regarding such a concept. In the following, it has been found relationships among fuzzy neutrosophic alpha generalized mapping as continuous, fuzzy neutrosophic mapping as continuous, fuzzy neutrosophic alpha mapping as continuous, fuzzy neutrosophic generalized semi mapping as continuous, fuzzy neutrosophic pre mapping as continuous and fuzzy neutrosophic γ mapping as continuous.

Our active aim in this paper is to prove the following Let Ŕ be a ring having an
idempotent element e(e  0,e 1) . Suppose that R is a subring of Ŕ which
satisfies:
(i) eR  R and Re  R .
(ii) xR  0 implies x  0 .
(iii ) eRx  0 implies x  0( and hence Rx  0 implies x  0) .
(iv) exeR(1 e)  0 implies exe  0 .
If D is a derivable map of R satisfying D(R )  R ;i, j 1,2. ij ij Then D is
additive. This extend Daif's result to the case R need not contain any non-zero
idempotent element.

It was known that every left (?,?) -derivation is a Jordan left (?,?) – derivation on ?-prime rings but the converse need not be true. In this paper we give conditions to the converse to be true.

The main purpose of this work is to introduce the concept of higher N-derivation and study this concept into 2-torsion free prime ring we proved that:Let R be a prime ring of char. 2, U be a Jordan ideal of R and be a higher N-derivation of R, then , for all u U , r R , n N .

In this paper, we prove that; Let M be a 2-torsion free semiprime  which satisfies the condition  for all  and α, β . Consider that  as an additive mapping such that  holds for all  and α , then T is a left and right centralizer.

In this paper, we study the concepts of generalized reverse derivation, Jordan
generalized reverse derivation and Jordan generalized triple reverse derivation on -
ring M. The aim of this paper is to prove that every Jordan generalized reverse
derivation of -ring M is generalized reverse derivation of M.

The main purpose of this work is to generalize Daif's result by introduceing the concept of Jordan (α β permuting 3-derivation on Lie ideal and generalize these result by introducing the concept of generalized Jordan (α β permuting 3-derivation 

In this paper, we investigate prime near – rings with two sided α-n-derivations
satisfying certain differential identities. Consequently, some well-known results
have been generalized. Moreover, an example proving the necessity of the primness
hypothesis is given.

Let M be a prime Γ-ring satisfying abc  abc for all a,b,cM and
,  with center Z, and U be a Lie (Jordan) ideal. A mapping d :M M
is called Γ- centralizing if u d u Z  [ , ( )] for all uU and  .In this paper
, we studied Lie and Jordan ideal in a prime Γ - ring M together with Γ -
centralizing derivations on U.

   Let M is a Г-ring. In this paper the concept of orthogonal symmetric higher bi-derivations on semiprime Г-ring is presented and studied and the relations of two symmetric higher bi-derivations on Г-ring are introduced.