[rrrrrrrrrrrr2]

I think this is an apples to oranges comparison. You get taught pemdas or bomdas early on and then use it hundreds of times throughout school and the rest of your life. The value of learning it is clear.

Whereas ring theory is very much specific to algebra and you're not going to use it programming, doing chemistry, or whatever else outside math. Finding motivation to learn pure math is a bit harder.

GP's example illustrates a good point. I agree with their point.