Which one is the correct way to write abstract algebra?
Let A be a Noetherian ring.
Let A be a noetherian ring.
I guess the cases Abelian/abelian, Artinian/artinian behaves similarly.
The word "abelian" is not traditionally capitalized in mathematics, perhaps because it has been generalized to absurdity from where it started. But generally, things named after people in the sciences are capitalized (see J.R.s answer for other exceptions). Reference:
Noetherian rings are named after Emmy Noether. I believe that when an adjective is formed from someone's name (or from some other proper noun), that adjective is generally capitalized, particularly in mathematics and science. That's why, in graph theory, Hamiltonian paths also use capital H, and why we spell Newtonian physics with a capital N.
In contrast, this is not done with the names of elements, even when the names are derived from proper nouns. For example, californium and einsteinium are spelled with lower-case letters, just like iron and hydrogen.