Reference on different types of integral domains

Question

I am looking for a good reference (book or otherwise) which has a comprehensive study of the different types of integral domains, including:

• Euclidean domains
• UFDs
• GCD domains
• PIDs
• Dedekind domains
• Bezout domains
• etc...

and especially the relationships between them (e.g. all Euclidean domains are PIDs, a Bezout domain is a PID iff it has the accp, etc). Thanks!

Answer 1

While I've never used a book that had all these topics at once, it does look like Bourbaki's commutative algebra volume might cover them:

• Bourbaki, Nicolas. Commutative algebra. Vol. 8. Hermann, 1972.

And Kaplansky may have a good deal of them:

• Kaplansky, Irving. Commutative rings. Allyn and Bacon, 1970.

Any good beginning ring theory book covers PIDs EDs and UFDs, namely

• Isaacs, I. Martin. Algebra: a graduate course. Vol. 100. American Mathematical Soc., 2009.

• Bhattacharya, Phani Bhushan, Surender Kumar Jain, and S. R. Nagpaul. Basic abstract algebra. Cambridge University Press, 1994.

• Grillet, Pierre Antoine. Abstract algebra. Vol. 242. Springer Science & Business Media, 2007.

I think Jacobson's Basic Algebra books contain at least UFDs and Dedekind domains.

For other more special types you'll probably do fine just looking at the references to articles given in the wikipedia articles.

As for information about their interrelationships there is a (not necessarily totally comprehensive) graph at the Database of Ring Theory for domains.