# Can polynomial rings be classified as a specific type of ring like algebraic integers are classified as Dedekind domains??

In algebraic number theory, we would like to study rings of algebraic integers but sometimes they are not PIDs and thus they don't possess good properties. Because of this, we have introduced the notion of Dedekind domains, in which rings of algebraic integer are included and one can study the nice properties of Dedekind domains to understand algebraic integers.

While doing algebraic geometry and commutative algebra, I have encountered polynomial rings in several variables, whose properties are quite different from polynomial rings in one variable, as the former ones are just UFDs while the latter one is a PID. So this "degeneracy" of polynomial rings from PID to UFDs reminds me of the similar case of rings of algebraic integers but I don't think they are related since the dimension of polynomial rings depends on the number of variables , which does not fit the definition of Dedekind domains.But I wonder if there is a way to classify polynomial rings of several variables(especially those over an algebraically closed field) into a specific type of rings, such that this type of rings has nice properties just like Dedekind domains and from which one can classify the prime ideals as well as determine the irreducibility of polynomials easily?

• The change from one to more-than-one "variable" is enormous... as it turns out. In terms of keywords, instead of "Dedekind", in several variables search on things like "Cohen-Macauley" or "excellent [rings]" or "complete intersection..." and many other things... Commented Nov 20, 2023 at 22:07
• One useful higher-dimensional generalization of Dedekind domains are Krull domains / rings. See also these graphs of common classes of integral domains. Commented Nov 20, 2023 at 22:07
• In addition to the above, here is a partial list of some standard properties for a multivariate polynomial ring over a countable field. There may be some differences for an uncountable field, and possibly some better things one can say for an algebraically closed field, but many will be the same. I would be happy to be informed of any differences people notice. Commented Nov 20, 2023 at 22:19
• @paulgarrett Macaulay. Commented Nov 21, 2023 at 11:42