This reminds me of the concept of a "ring of functions" (albeit slightly more constrained). Since you seem to be familiar with measure theory already, that idea is probably already well known to you (the author), but I figured I'd mention it anyway.
I didn't know about that before. Looks very interesting (looking at wikipedia) — thanks for mentioning it!
I highly recommend learning abstract algebra thoroughly if you haven't already, by the way. It's a lot more well connected and well researched than you might think after just reading about this or that definition (groups, rings, quotient rings, lattice of ideals) in a vacuum. The magic comes when you mix all the definitions together. If you learn it correctly and master both the rigour and intuition behind it all, you'll come to consider your pre-abstract-algebra-self in the same Platonic cave as you might see your pre-linear-algebra-self right now.