The aims of this paper are introduce the notions of symmetric bi-derivation and generalized derivation in M V-algebras and investigate some of their properties.

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The purpose of this paper is to prove some results concerning permuting triderivations and permuting generalized triderivations on prime and semiprime rings which partially extend some results contained in [9] and [10].

In this study, we investigate the problem of how to define [Formula: see text]-nearness semiring in the sense of Nobusawa theory which extends the notion of a nearness rings, nearness semirings and [Formula: see text]-rings [N. Nobusawa, On a generalization of the ring theory, Osaka Journal of Mathematics 1 (1964) 81–89; M. A. Öztürk, Semirings on weak nearness approximation spaces, Annals of Fuzzy Mathematics and Informatics15(3) (2018) 227–241; M. A. Öztürk and E. İnan, Nearness rings, Annals of Fuzzy Mathematics and Informatics 17(2) (2019) 115–132]. Also, we introduce some properties of approximations and their algebraic structures.

In the paper we examined the some effects of derivation, trace of permuting tri-derivation and endomorphism on each other in prime and semiprime ring.Let $R$ be a $2,3$-torsion free prime ring and $F:R\times R\times R\rightarrow R$ be a permuting tri-derivation with trace $f$, $ d:R\rightarrow R$ be a derivation. In particular, the following assertions have been proved:1) if $[d(r),r]=f(r)$ for all $r\in R$, then $R$ is commutative or $d=0$ (Theorem 1);\ 2) if $g:R\rightarrow R$ is an endomorphism such that $F(d(r),r,r)=g(r)$ for all $r\in R$, then $F=0$ or $d=0$ (Theorem 2); 3) if $F(d(r),r,r)=f(r)$ for all $r\in R$, then $(i)$ $F=0$ or $d=0$, $(ii)$ $d(r)\circ f(r)=0$ for all $r\in R$ (Theorem 3). In the other hand, if there exist permuting tri-derivations $F_{1},F_{2}:R\times R\times R\rightarrow R$ such that $F_{1}(f_{2}(r),r,r)=f_{1}(r)$ for all $r\in R$, where $f_{1}$ and $%f_{2}$ are traces of $F_{1}$ and $F_{2}$, respectively, then $(i)$ $F_{1}=0$ or $F_{2}=0$, $(ii)$ $f_{1}(r)\circ f_{2}(r)=0$ for all $r\in R$ (Theorem 4).