Question

a second-ish course on algebra

Answer 1

I recommend Pierre Antoine Grillet - Abstract Algebra and for Module Theory, Thomas J Head - Modules: A primer of structure theorems or Thomas Scott Blyth - Module theory: An approach to linear algebra.

Answer 2

I've enjoyed Hungerford's Algebra text, one of the graduate texts in Mathematics books.

Answer 3

If you want something comprehensive, Hungerford is probably your best bet. It's normally used as an intro-level graduate text, so it might not be what you're looking for, but it goes deep enough that there will be plenty of new material and the exercises are very good.

Answer 4

I think a good basic algebra text with nice problems is "Algebra" by Godement. Although I don't know D & F well, my impression is that the exercises are at a higher level in Godement. However, Godement stops short of Galois theory and doesn't try to be comprehensive the way D & F seems to.

At a higher level, with harder problems, is Basic Algebra I by Jacobson. Whether to recommend this at this stage sort of depends on whether you think the reason you did badly was something temporary like a lack of time or if you think you have genuine difficulties with algebra. I think it meets the definition of a "second course" though. The third chapter (of eight) is on finitely generated modules over a PID and is really great, and the fourth chapter is on Galois theory.

If you can read French, an easier follow-up to Godement would be "Cours d'algèbre" by Perrin, and there are a couple of books by Malliavin covering topics including representation theory, commutative algebra and Galois theory.