Does the prime modulus have to be bigger that the generator?

Question

In the DH algorithm $ a^x \bmod b $ where $ a $ is the generator and $b $ is the prime modulus does $b $ have to be bigger than $ a $ or could $ a $ be larger for example $ 19^x \bmod 17 $ . Because I don't see a problem with the generator being larger because the answer can still only be $ b \ge \text{answer} > 0 $ . Obviously the prime modulus would still have to be large enough to provide good security but I was just curious if the generator could be larger.

Answer 1

I suppose there is really no requirement to have $a <b$. But then again, if you are using an $a>b$ why not reduce it modulo $b$ and save space?