Undergraduate] Ring Theory: which books have detailed worked-out examples?
I’m taking an undergraduate Ring Theory course and we’re currently using Gallian's Contemporary Abstract Algebra. I really like Gallian, but I’d like to supplement it with another book that has more detailed proofs, worked-out examples, and explanations of how to approach problems.
Our syllabus is roughly:
Unit I – Rings and Ideals
Definition and examples of rings
Properties of rings and subrings
Integral domains and fields
Characteristic of a ring
Ideals and operations on ideals
Ideals generated by a set
Factor rings
Prime and maximal ideals
Principal ideal domains
Unit II – Ring Homomorphisms and Polynomial Rings
Ring homomorphisms
First, second and third isomorphism theorems
Field of quotients
Polynomial rings over commutative rings
Division algorithm and its consequences
Unit III – UFDs and Divisibility in Integral Domains
Factorization of polynomials
Reducibility tests
Mod p irreducibility test
Eisenstein's criterion
Unique factorization in ℤ[x]
Divisibility in integral domains
Irreducibles and primes
Unique factorization domains
Euclidean domains
I’m particularly looking for a book that doesn’t skip too many steps in proofs and has plenty of worked examples before giving exercises.
For someone at the undergraduate level, which would you recommend as the best supplement to Gallian?
Thanks!