I am going to be taking a first course in abstract algebra this fall and we are using Jacobson's Basic Algebra I for the course text. It looks like a good textbook, but it is lacking a lot of motivation (I am new to higher math), so I was wondering if there are other books to supplement my studies. Particularly historical or geometric motivations; Something to give me an idea about why we are learning about ideals, modules, etc. so I can "picture" what we are learning.
Thanks for any input!
Best Answer
That's a good question that - alas - does not really have a good answer - at least not a single textbook that one could point to. The best motivation for ideals and modules comes from number theory. As such, you might find it helpful to attempt to learn some algebraic number theory in parallel, e.g. factorization theory in quadratic number fields. Unfortunately most of the algebraic number theory textbooks already assume solid algebra background, so you'll need to interpolate your own motivational examples.
One crucial point deserves emphasis. When first learning about new abstractions such as ideals and modules it is absolutely essential to understand their motivating concrete instances. Thus when you are learning about ideals in algebra you should revisit your knowledge of elementary number theory and reinterpret results in terms of these new abstractions. For example, see my post here and its links - which explains how various proofs of irrationality of sqrt's by descent on the size of a denominator are merely special cases of proofs that a denominator ideal - Dedekind's conductor ideal - is principal or invertible. From this general viewpoint the proof is a one-liner: the conductor ideal is invertible so cancelable. This shows that any Dedekind domain or PID is integrally closed - a monic generalization of the rational root test to a very wide class of rings. This is but one of many generalizations that you will meet when studying factorization theory in wider classes of rings. To better hone your intuition it is essential that you learn to easily move back-and-forth in this way between the concrete and abstract. Unfortunately most textbooks omit explicit discussion of such motivational material, so you'll need to develop it on-the-fly as need be.