Any standard abstract algebra textbook will cover rings in gross detail. Dummit and Foote is a much beloved standard, but is also used in graduate classes. I've heard good things about Pinter's book (http://www2.math.umd.edu/%7Ejcohen/402/Pinter%20Algebra.pdf).

For lattices, there's Birkhoff's book on lattice theory, which is where I learned what I know about it. I haven't spent any time with other books.

Really enjoyed Algebra: Category 0. Introduces some basic Category Theory and uses it to develop the standard practice of undergrad algebra. The exercises are medium difficulty (nothing ridiculous like a Knuth or Spivak problem, but nothing breezy either) and there are enough examples in the text of proofs with enough detail to get you working on the exercise problems.

Did you mean Aluffi’s Algebra: Chapter 0? What a wonderful book.

Sorry yes

Herstein Topics in Algebra or Dummit and Foote.

goofy suggestion. those are both either grad or senior undergrad books depending on where you are. much better suggestion is

A Book of Abstract Algebra by Pinter (really gentle)

or

A First Course in Abstract Algebra by Fraleigh

I used on in high school and the other my second year of undergrad. They have lots of material but they also have good exercises and explanations.

Herstein was my undergrad book back in the early 90's. I remember really liking it. It's been a very long time, but my recollection was that the groups to rings progression felt a lot more natural to me than the rings to groups progression of some of the other books at the time.

Dummit&Foote imho is the best undergrad algebra book.

Lattices (the algebraic structure) seem to exist only on Wikipedia in the sense that the only time I've seen lattices every mentioned in my standard undergrad education is once for a proof of the Stone-Weierstrauss theorem.

I know there are textbooks on the topic, and probably lots of people who deal with lattices a lot. But my own experience seems to be that wikipedia puts more emphasis on lattices and things like universal algebra than actually happens in math.

There is an interesting rant from Gian-Carlo Rota on why lattices are so rarely taught despite being so ubiquitous. I include them based on the amount of use I have gotten out of them over the years. For example, the entire theoretical structure of eventual consistency is "make your merge operation the meet of a semilattice."