>>14358908Commutative algebra and the study of rings and modules is just a convoluted way of doing elementary number theory. Most of the objects and properties you deal with are either generalizations of prime numbers, factorization, and division algorithms, or consequences of these sorts of generalizations. Youre just studying ordinary factorization, but ideals let you talk about collections of integers and how they factor, rather than just individual integers. Its basically a change of persoective from individual numbers to collection of numbers, with a bunch of nee terminology and notation that is largely unnecessary.
Commutative algebra is only useful insofar as it serves as preparation for a the study of certain topics concerning matrices and algebraic geometry.
Non-commutative (and especially non-associative) algebra, on the other hand is very interesting, because we know very little about these topic, and because pretty much anything intersting in physics or computer science involves some sort of non-commutative or non-associative object. E.g. quarternions in physics and strings with amiguous parse trees in computer science are both non-commutative, and the latter is even non-associative.
