So I've recently gotten interested in learning some math again after a long hiatus from the courses I took in school. The subject of abstract algebra in particular fascinates me, and I've read about how things like groups, rings and fields are elegant objects that generalize a lot of concepts and form the foundation for a lot of higher mathematics.

My long-term goal is to be able to gain a solid and thorough understanding of the concepts covered in a book such as the one by Dummit and Foote (which I've heard seems to be the classical text on abstract algebra). I'm not taking a class and I don't have an instructor, so all of my help will have to be from textbooks and online resources.

I know that I won't be able to start a book as advanced as the one above anytime soon, and will have to progress up to it. I started reading the fourth edition of "Mathematical Proofs: A Transition to Advanced Mathematics" by Chartrand, Polimeni and Zhang so I can first get comfortable writing proofs. I'm still only on the chapter about sets, but I'm finding that it's progressing at the perfect pace and right difficulty level for me so far. I've done the exercises in the first few sections and have gotten most of the answers correct.

However, since it's been so long since I've taken a math course, I'm worried that I may have forgotten some of my previous math knowledge and may eventually hit a barrier the further I go. I took courses in Calculus I and Linear Algebra, and did well in both of them, but they were many years ago. The Calculus course covered differentiation and its applications, but not integration or functions with several variables. The Linear Algebra course was mainly about matrices and vector spaces. It's been a while since I've seen the concepts from these courses. Even in Pre-Calculus, I'm finding that I can't remember off hand how to simplify a rational expression or find a trigonometric function. Hopefully, when I get reacquainted with these things, they should quickly come back to me.

So I ask: Do you think it's a good idea that I continue with the book that I'm reading about proofs, and just refer back to notes in Calculus and Linear Algebra once I stumble upon them as I go? Or do you think I should start back at the beginning and relearn everything? How well do I have to be in these two subjects to be able to get a good grasp of Abstract Algebra? Would learning the concepts in Calculus II and III and in more advanced Linear Algebra be necessary?

After the book I'm reading (which, by the way, has two chapters introducing groups and rings) I thought I might be able to progress into a proper text about Abstract Algebra, like the one by Fraleigh (I've quickly skimmed through it and it seems to have a much more gentle introduction than the one by Dummit and Foote). Do you think this is a good choice? Any other books, resources or tips do you think I should be made aware of?

Thanks so much.
Calculus can be a source of examples in abstract algebra (rings of differentiable functions, differentiation as a derivation), but logically the subjects are pretty independent. Linear algebra is much more relevant, and many of the most fundamental examples in abstract algebra come from linear algebra. Having said that, you could learn group theory without knowing linear algebra, but for rings, fields, and modules, I would definitely try to study some linear algebra first.
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>I'm worried that I may have forgotten some of my previous math knowledge and may eventually hit a barrier the further I go.

All of us forget previous knowledge and hit barriers. That's just part of the deal when studying math. I think as long as you are having a good time there's no need to worry :)
Interesting post, as I am going through the exact same...

Graduated with a master's in Astrophysics, and although I kept up with the general astronomy knowledge, the math portion has been ignored in-toto.

But recently I realized that I am enough interested in specific subjects (stellar evolution in particular) to dive back in, and just like you hit a brick-wall-of-math.

So, my approach has been to more-or-less follow the same route as what I went through in grad school: pick up the calculus book, and at the same time start with an easy applied course, in my case: Statistical Physics. And this way I hope to work my way back into stellar evolution and stellar atmospheres.

One thing I find is that I have matured, and that what I found impossible to understand in calculus I now have the patience to work through. And hopefully, by combining it with a more applied subject, Ill stay interested enough in the math to persist
D&F is great and I use it often for reference, but I found Fraleigh to be the easier self-study. I ultimately learned most my algebra from Fraleigh.

If you've covered calculus/lin alg before, and have *some* memory about how they work, you're good. You'll understand the examples that use them. If you don't, just revisit that topic.

The bigger struggle will be getting used to a proof-based framework, which you've already gotten a great head-start on. This is a hurdle for many students.
How’s your set theory ?  Read that little book Naive Set Theory. If you canget through it and do a lot of exercises (not all, say 40%) you are ready.
I'll just answer the last question since it's more opinion based. Dummit and Foote is one of my favorite books, but it's way too expansive and detailed to self-study.

I personally don't like Fraleigh. Something on a similar level of D/F is Herstein's Topics in Algebra. It's much more concise, and while some of the later parts aren't amazing, the group theory section is great. If you want something gentler, I'd recommend Pinter's algebra book. It has plenty of simple exercises and walks you through a lot of important ideas in the exercises.
As an algebraist, I'd say some foundational learning in discrete mathematics would be a good background for abstract algebra. If you have the basics covered (permutations, set theory, etc.) I'd say you can start on an abstract text (though the one by D&F, as noted above, may be a little too dense for practical self-study). For linear algebra, if you want to refresh and likely advance your background I recommend Axler's "Linear Algebra Done Right" as a jumping-in point, but is not necessary for your study unless you come across terms that trip you up. I recommend using it as more of a reference than study material if you are purely interested in objects such as Groups, Rings, and Fields. Many examples of such objects relate back to foundational courses in calc and linear algebra, so while you dont need to study these things specifically it may help with comprehension. For Abstract Algebra texts I like Nicholson.