The best undergrad algebra textbook I've studied is
Algebra by Mac Lane and Birkhoff (3rd edition! the previous editions aren't quite as good and substantially different; i haven't seen the 4th edition and it is out of print so /shrug). I've used multiple books both in self-study and class and this book is, to me, in a league of its own. Not only does
Algebra teach modern algebra, it teaches one to think like a modern algebraist, and not like just any modern algebraist, but like Saunders Mac Lane who was pretty great at algebra.
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.
[1] https://en.wikipedia.org/wiki/Isomorphism_theorem
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.
[1] https://en.wikipedia.org/wiki/Isomorphism_theorem