Abstract
Let R be a ring. A map F:R -> R is called a multiplicative (generalized)-derivation if F(xy) = F(x)y + xg(y) is fulfilled for all x, y is an element of R, where g:R -> R is any map (not necessarily a derivation). The main purpose of this paper is to study the following situations: (1) F([x, y]) = +/-(xy +/- yx), (2) F(x circle y) = +/-(xy +/- yx), (3) [F(x), y] +/- [x, G(y)] = 0, (4) F(uv) +/- uv = 0, (5) F(uv) +/- vu = 0, (6) F(u) F(v) +/- uv = 0, (7) F(u) F(v) +/- vu = 0, (8) F(uv) = F(u) F(v), (9) F(uv) = F(v) F(u) for all x, y, u, v in some appropriate subsets of R.