Thank you very much, especially for the links. Could you introduce me some more well-grounded references that the notions of (regular, Gorenstein and C.M.) and their interactions have been explained more with sufficient details? I'm really interested in learning more about them!

Great! that is a fruitful explanation. Thanks.

How do you deal with the equation $grade(I)=ht(I)$ for any ideal $I$ of $T$?

Dustin Cartwright: Thanks for your complete answer. That is a careful point.

@ACL: Good point. Is it possible to understand $Spec(R)$ in this example? The point is that $P(A)$ where $A \subset \mathbb{N}$ is an ideal but not all ideals are being as such!

For 1) yes, definitely, I meant a consistent set of equations. For the first line of my question I intended to give the background in a few words.