I am very interested in ring theory. I want to know the best textbooks about that subject, i.e about PID's, UFD's, Dedekind domain, Euclid domain. Can anyone tell me?
It would be a good choice to go for Abstract algebra text by Dummit Foote for the part which you have asked for .
Simultaneously Keep doing problems (*in the same order as it is given)
Do not take it much seriously but it would be a good idea if you can try looking at "Algebra : Michael Artin" mainly for irreducible polynomials part.
Simultaneously, you please keep a copy(*soft) of "CONTEMPORARY ABSTRACT ALGEBRA-Joseph Gallian". This have only small portion in chapter "Divisibility in Integral Domains" but i would say this is very useful.
Caution : My idea of suggesting three books is not to make you get confused but just for reference and one more use is something which is asked as an exercise in one book is proved as lemma in another book :P which would be useful.
I find the book "P.M.Cohn: Introduction to Ring Theory" very useful. For noncommutative rings
there is the book "T.Y.Lam : A First Course in Noncommutative Rings". The level is a little higher than that of the first reference, but the book is not difficult. Good luck! (I must say I am pleased that you are interested in ring theory. I find the area very rich and exciting.)