Most of these rings are a combination of rough casting (the bits under the burner caps) and a polished and anodised finish.
As far as I can tell (I detail cars, clean anodising on motorcycles and have an M.Eng), there is nothing that will get burned on splashes off without damaging/removing the anodising.
Anodising looks great and can be used to give those coloured finishes on aluminium bits (e.g. metal plugs on charging cables). But it is sensitive to a lot of chemicals and not very tough.
If you have ever had a shiny aluminium coffee maker (Bialetti etc), that is anodised.
They will go dull if you clean even with washing up liquid.
By "tough" (see above), I mean that if you use an abrasive, it will remove the anodising easily. You then have bare aluminium.
Bare aluminium goes full because it forms a layer of aluminium oxide in reaction with the air quite quickly.
That's why shiny unpainted alloy wheels have to be coated with clear lacquer.
The marks can be cleaned off in my experience with a green scourer, but you WILL remove the anodising as well.
It's basically a completely unsuitable finish and the manufacturers know this but can still sell hobs so they don't care.
Same as the stainless hobs (enameled was always better, but people like the look of a new stainless hob and it's cheaper to make) and the lettering/markings on stainless steel hobs which typically wears off unless you only use plain water to clean (which won't work if course).
The only answer I've found is to abrasively remove deposits using the least bad way you can. Try green scourer with plenty of suds to lubricate it.
Then put up with dull aluminium. If you want it shiny, you can polish the aluminium and it will last a while.
Lots of metal polishes around. I use Autosol as I already have that in the garage. It's quite quick and easy. Use one cloth to put the polish on and then another to polish off (this works much much faster and better than a single cloth).
And review products negatively, buy something different next time. The manufacturers will get the message soon enough.
P.s. I know this is an old thread, but it's the one I found Googling to see if anyone has found a better solution.