Some identities involving multiplicative generalized derivations in orime and semiprime rings

Let $R$ be a ring with center $Z(R)$. A mapping $F:R\rightarrow R$ is called a multiplicative generalized derivation, if $F(xy)=F(x)y+xg(y)$ is fulfilled for all $x,y\in R$, where $g:R\rightarrow R$ is a derivation. In the present paper, our main object is to study the situations: (1) $F(xy)- F(x)F(y)\in Z(R)$, (2) $F(xy)+ F(x)F(y)\in Z(R)$, (3) $F(xy)- F(y)F(x)\in Z(R)$, (4) $F(xy)+ F(y)F(x)\in Z(R)$, (5) $F(xy)- g(y)F(x)\in Z(R)$; for all $x,y$ in some suitable subset of $R$