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.