Advanced Algebra textbooks
math.stonybrook.edu
math.stonybrook.edu
My first AA book was the "European kind", all symbols and definitions, a few proofs every ten pages. It was too dry for me. I never thought other people would think that way.
I love brain teasing but I also need a minuscule amount of inspiration to power my neurons.
For an amusing instance of induction on dimension, you might enjoy the proof of the AMGM inequality, which proceeds by upwards induction that doubles the dimension, followed by downward induction: https://proofwiki.org/wiki/Cauchy's_Mean_Theorem#Theorem .
Skip the proofs on the first read (this is the implementation, and may or may not be enlightening.)
But, number one rule with learning maths is: you got to do it yourself. Play with it somehow. It's similar to learning a new (or first) programming language (or API): have a project in mind and try to do it using that language.
Seriously, you absolutely cannot learn this stuff just by reading. Or, at best you may learn a very small fraction of it.
IMO, this text is far from "typical definition-theorem-proof". There is plenty of other prose and examples there aswell.
what would be great is if typesetting tools improved sufficiently so that one could choose 'beginner' or 'advanced' mode when reading a maths textbook. perhaps that is too fanciful!
As an example of Algebra's approach, take the isomorphism theorems [1]. Now many undergraduate textbooks (like Dummit and Foote) will prove these theorems by manipulating cosets and deal with gross "implementation details" at the level of sets. Mac Lane insists otherwise: The only time you have to manipulate cosets is in order to construct the quotient G/N of a group G by one of its normal subgroups N. Once you have constructed this group and proved its universal property, the isomorphism theorems can be proved without ever mentioning cosets again. What is that universal property? It has two parts: First is that there is a morphism p from G to G/N which sends all of N to the identity in G/N. Second is that any morphism f from G to any group L that sends all of N to the identity in L necessarily factors uniquely up to isomorphism as a composition of morphisms g ∘ p. This is the essence of a quotient group.
Mac Lane's approach is to apprehend the essence of what is studied while discarding as much of the set theoretic husk as is possible. It is algebra in its purest form, accessible to and transformative of the mind of an undergraduate. Reading this book is a recurring joy to me.
"Basic Algebra" means "material typically covered in late middle or early high school".
Then the word "abstract" should appear in the title.
The material here is precisely what you'd expect in your first (i.e., basic) college algebra class.
In terms of being easier to read, I find the extreme contrast between thick and thin strokes does not work well on a screen.