The Ring of Protection gives a deflection bonus, which does stack with normal armor bonus. It does not, however, stack with other deflection bonuses. So:
Armor + Shield + Ring of Prot. = A-Ok
But multiple Rings of Protection will not stack. So, yes, you are right and the GM is wrong.
Show him your character sheet where the boxes of ac bonuses are. There is a box for deflection bonus, and a box for armor bonus, and a box for the total. This is because they are different types of bonuses and therefore stack.