non-linear new product abpi derivations on rings

Let r be aunital prime ring with unit i which contains a non-trivial idempotent p_1. we will show that anyarbitrary not necessarily linearmap φ:r→rsatisfies the conditionφ(abp_i )=φ(a)bp_i+aφ(b) p_i+abφ(p_i ). i=1.2 (0.1)for alla.b∈r, where p_2=i-p_1, is an additive derivation.let r be aunital prime ring with unit i which contains a non-trivial idempotent p_1. we will show that anyarbitrary not necessarily linearmap φ:r→rsatisfies the conditionφ(abp_i )=φ(a)bp_i+aφ(b) p_i+abφ(p_i ). i=1.2 (0.1)for alla.b∈r, where p_2=i-p_1, is an additive derivation.