DERIVABLE MAPPINGS AND COMMUTATIVITY OF ASSOCIATIVE RINGS

Abstract

Let R be a ring with center Z(R).  A mapping F : R \(\longrightarrow\) R (not necessarily additive) is called a multiplicative (generalized)-derivation of R if it is uniquely determined by a mapping d : R \(\longrightarrow\) R such that F(xy) = F(x)y + xd(y) for each x, y ∈ R.  In the present paper, we investigate the commutativity of a semiprime (prime) ring via studying a number polynomial constraints involving multiplicative (generalized)-derivations.  Moreover, some annihilator conditions are also examined.