One important starting point that is familiar to even high school algebra students: Integers and rings of Polynomials (say, the ring of polynomials with integer coefficients) are examples of rings. They have a lot in common -- we see very similar ideas of factoring, division with remainder and related theorems (uniqueness of factorization, the Chinese Remainder Theorem etc.). So while this sort of Unique Factorization Domain isn't the only sort of ring we study, coming up with something that generalizes this sort of behavior should make sense.

Functions on any space form a (commutative) ring (by pointwise addition and multiplication).

This is usually a duality (from the ring you can reconstruct the space). This is the fundamental starting point of algebraic geometry (but it also works with topological spaces and continuous functions, etc).

So you could say that (commutative) ring theory is the study of spaces.

Whenever you have an abelian group A, the group homomorphisms A->A naturally form a ring, with multiplication given by the composition. This ring encodes many important structures of the original group A, while having an additional piece of structure makes it more well behaved

basically Z we want to have facts about Z proven more generally additionally attempts to prove Fermats Last Theorem historically.

Here is a story on the historical motivation for the study of rings. The mathematicians Gabriel Lamé and Ernst Kummer both independently discovered a false proof of Fermat's Last Theorem in which they assume Z[r] where r is a root of unity is always unique factorization domain (UFD). It's now known that if r is a 23rd root of unity, for example, then Z[r] is not a UFD.

This failure of unique factorization led Kummer to a study of what he called "ideal numbers." It's not very clear to me what exactly ideal numbers are, but I do know Kummer's work led to Dedekind's definition of an ideal in a ring, and the formal study of rings more generally.

It also should be noted that the individual pieces of this story are correct, but their connection has been disputed. We don't really know how motivated Kummer was by FLT specifically, and it's often exaggerated that his later work on ideal numbers was due to him being so bothered by the failure of this FLT proof.

As a category theorist, I would say Rings are more fundamental. The category Ring (of unital rings) can be constructed as the category MonAb, the category of monoids over abelian groups (the underlying abelian group of a ring is the additive group). Then, the integers appear as the initial object of Ring. (PS: The additive group of integers is significant in that it is the free abelian group generated by one element.)

"God gave us the integers, the rest is the work of man" -Kronecker

you already know that the integers are the initial object in commutative rings with unit (let's call it: CRing_1) .. so it comes down to what you find "fundamental"? I'd argue, integers are a consequence of the category CRing_1, but you couldn't reconstruct CRing_1 just from staring at the integers long enough imho.

specifically the standard proof of the fact that determinants "stay multiplicative" over commutative rings, where one argues over Z and its quotient fields for various numbers of variables .. seems to convey the integers "know everything", but I'd argue they know some things, the ones every commutative ring must at least obey..

and then I go into the category of spectra, where the sphere spectrum is initial, and can "point to" the (eilenberg mac lane spectrum of) the integers .. suddenly you could consider the "field with one element" .. maybe spectra are _the_ fundamental object? :)

integers are more fundamental because we found them first :)

I understand Z-modules are the same as abelian groups, but the second notion feels more fundamental. How do you even define a module without Ab?

Ian Stewart on Galois theory.

There's a book by Judson that is free online. https://judsonbooks.org/aata-files/aata-html/aata.html

https://judsonbooks.org/aata/

> and I need a text book

since you're in university, try the university bookstore

Dummit and Foote

https://3lihandam69.wordpress.com/wp-content/uploads/2020/02/a-first-course-in-abstract-algebra-jb-fraleigh-7ed2003-new.pdf

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Life pro-tip: every book becomes highly affordable when you use library genesis. But of course I’d never recommend something like that.

Abstract Algebra by Durbin gets recommended by my university. Not brilliant but it likely covers what you’re looking for

I study better with a physical book so I recommend A Book of Abstract Algebra by Pinter

Judson's text is available free online: http://abstract.ups.edu/download.html

As this is basic abstract algebra, there is no need to get a book for that. Lecture notes would do equally well. For instance, look at

https://math.berkeley.edu/~apaulin/AbstractAlgebra.pdf

If you want to go further than that, just let me know.

Hungerford.

Kidding, of course. What I really meant was Hungerford. He has two books. Try the introductory one.

Another good one is Gallian.

Late undergrad to beginning grad you can try Papa Hungerford, Dummit and Foote, Aluffi, and Lang.

I spent a minute thinking about it, but nothing ideal came to mind -- every kernel of an idea just hints at a silly pun or something long and barely comprehensible (e.g. "jokes" about composite algebraic structures).

Hopefully you find some closure or decide on a pretty but meaningful graphical representation.

Hey girl, we must have a repeated root in some field extension because we are inseparable

I'm sorry I will leave now...

Our relationship is not Noetherian because my love for you will never stabilize

There's gotta be something in here... https://www.youtube.com/watch?v=BipvGD-LCjU

I don't have something for your rings, but one of my favorite jokes is that you can tell Beyonce is a commutative geometer because when she sees something, likes it and puts a ring on it.

I just want to say, regarding mathematics textbooks generally; the author has to economize space and their time spent working. The book needs to be of a reasonable size at a reasonable printing cost and the author usually also has time constraints. Also, the point of the textbook is not to spoon feed everything to you, but to get you through the typically challenging areas and intentionally leaving the rest for the reader to puzzle out for themselves. Ultimately, you have to do math to learn math; you can't absorb it from passive reading and osmosis.

Presumably the book did spend time showing (or else took axioms or definitions assuring) we have closure under addition and multiplication. It is an obvious consequence of those things that any polynomial in integers is going to be an integer, such that the reader should be expected to be able to do it themselves.

The validity of, say, the Euclidean division algorithm may seem obvious to someone who is used to working with integers, but that it follows from certain other properties of the integers really is not, at least not to someone who doesn’t already have background in commutative algebra. Why does the Euclidean division algorithm work in Z and Z[i], but not in Z[sqrt(5)i]? Is this something that should be explained? How many people reading an introductory book in number theory can prove the Euclidean division algorithm works from given axioms, and identify exactly which axioms are necessary for this result?

This problem isn't unique to number theory. Knowing what needs justification vs what is obvious enough to accept is a difficult question to answer. It depends on a lot of things: the field, the audience, the author's preference, etc. It's the kind of thing you do have to pick up as you go. One rule of thumb I use is if I can construct the entire proof in my head with little to no effort, I don't bother writing it down. This doesn't always apply, but it's a starting point.

There's no time to explain; come with me if you want to live prove interesting theorems.

Because otherwise it would be a really long book.

This may depend on two other things: 1) What are your background levels in connected other areas (e.g. topology, analysis and number theory) and 2) what aspects of group theory and ring theory are you most interested in?

If you want something that ties in with a lot of really cool geometry that you might see soon, I'd recommend "Abel's Theorem in problems and solutions". Links Riemann surfaces and group theory.

If you are simply looking for harder: Jacobson or Lang are the two id recommend.

If you are looking for a different approach but around same difficulty: allufi