In a ring, the characteristic is the smallest integer n such that nx=0 for all x in the ring. Is it acceptable to take "f" of both sides to get: f(nx)=f(0) in the corresponding polynomial ring? If so,...

In a ring, the characteristic is the smallest integer n such that nx=0 for all x in the ring.  Is it acceptable to take "f" of both sides to get:

f(nx)=f(0) in the corresponding polynomial ring?

If so, is f(0) 0 in the polynomial ring?

And can we write f(nx) as nf(x)?