Lie maps on alternative rings

Let R be an alternative ring containing a nontrivial idempotent and R ' be another alternative ring. Suppose that a bijective mapping phi:R -> R ' is a Lie multiplicative mapping and D is a Lie triple derivable multiplicative mapping from R into R Under a mild condition on Rwe prove that phi and D are almost additive, that is, phi(a+b)-phi(a)-phi(b)is an element of Z(R ')and D(a+b)-D(a)-D(b)is an element of Z(R)for all a,b is an element of Rwhere Z(R ')is the commutative centre of Rand Z(R)is the commutative centre of R As applications, we show that every Lie multiplicative bijective mapping and Lie triple derivable multiplicative mapping on prime alternative rings are almost additive.