A note on linear derivations

At first we prove some results on a general polynomial derivation using few results of linear derivation. Then we study the ring of constants of a linear derivation for some rings. We know that any linear derivation is a nonsimple derivation. In the last section we find the smallest integer w > 1 such that the polynomial ring in n variables is w-differentially simple, all w derivations are nonsimple and the w derivations set contains a linear derivation.