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.