I would say it all comes down to how soon you want to start on Tear of the Goddess. Some people think if you don't rush it, you will hurt yourself in the long run. With Ryze, I can kinda see their point since the only spell he can cast without a target is his Q. The cooldown without resetting it with W and E is long enough that stacking can be difficult. I also think that his clear efficiency actually makes stacking Tear kinda difficult if you're going for Archangle's. Manamume is a bit easier because you can stack with auto attacks.
Delaying Tear with Doran's Ring can still be completely viable however. It is a better starting item because it offers more utility without needing to back very early for that Tear upgrade. Sticking in lane for the full 750 gold may be more efficient since it becomes easier to farm with additional AP
I have another start for you to try though. I am not going to promise that it is any better, again it really just comes down to personal preference. Start with Dark Seal and Refillable Potion. Dark Seal will allow you to regenerate more health, more quickly with each charge of your Potion, than Doran's Ring offers in flat health, and gives a flat 100 mana for Ryze to work with. It's AP starts out the same as Doran's Ring, and stacks with Glory granted from kills and assists. If you can hang in lane for 1100 gold, grab your Tear and upgrade your Refillable Potion into a Corrupting Potion. This makes for very aggressive trades. Pop a charge when you engage and you will almost certainly out trade just about anyone.