Does Bezout's Identity hold for Zero cases?

Question

In some places I see Bezout's Identity stated for any two non-zero numbers $a$ and $b$. In other places it is stated that $a$ and $b$ are not both zero (so one of them can be). But doesn't Bezout's Identity hold for $a$ and $b$ both being zero, where it is assumed (as I see often done) that $gcd(0,0) = 0$? And if so, what is the reason for the non-zero constraint on one or both of the numbers?