Over the past 3( 9 sessions) weeks my professor has covered entire Part 3 - Rings from Gallian's Abstract Algebra which includes
Introduction to Rings
Motivation and Definition Examples of Rings
Properties of Rings
Integral Domains Definition and Examples Ring
Ideals Factor Rings Prime Ideals and Maximal Ideals Characteristic of a Field
Ring Homomorphisms Properties of Ring Homomorphisms
Polynomial Rings Division Algorithm Consequences of Division Algorithm
Reducibility Tests Irreducibility Tests Unique Factorization in Z[x]
Historical Discussion of Fermat’s Last Theorem Unique Factorization Domains Euclidean Domains
￼ ￼All this included 100's of definitions and proofs with what seems like has no rhyme or reason.
For example, Ideal is just introduced with this definition
A subring A of a ring R is called a (two-sided) ideal of R if for every r in R and every a in A both ra and ar are in A.
And the author goes on introducing Ideal tests, Maximal/ prime ideals in his usual definition proof example format followed by hundreds of exercises.
Concept after concept, proof after proof without motivation or any sort background as to why we need an Ideal, how it came into being.This to me is madness.
I got absolutely repulsed by that textbook and picked up Artin's Algebra book he introduces isomorphisms , kernels and Ideals
The property of the kernel of a ring homomorphism-that it is closed under multiplication by arbitrary elements of the ring-is abstracted in the concept of an ideal. This peculiar term "ideal" is an abbreviation of "ideal element," which was formerly used in number theory. We will see in Chapter 11 how the term arose.
This to me made more sense than pulling out a definition out of thin air.
I feel angry that I wasted a semester memorizing definitions and working out proofs based on those definitions without actual understanding of motivations, history etc. I also have a feeling that I will promptly forget all the ideas in a month or two because I didn't connect them in my head to existing concepts that I knew ( that's how we learn, right? ).
My question is How do I prevent this from happening to me in future?
Is math supposed be learnt like you are learning a game where you are given all the rules ( definitions) and you have to play the game according to those rules( theorems, exercises) ?
why are textbooks like Gallian's popular in math instruction? How are you supposed to read them? Why do professors go through proof after proof with no rhyme or reason?