>>640
Thank you for your response.
At first, I would like to ask the difference between algebraic closed fields and general commutative fields in terms of ring theory.
When we treat finite dimensional algebras over a field $k$ as $k$-modules,
I guess, we just have the property that these algebras are free as $k$-modules.
This is my trouble.
If so, I wonder why we treat finite dimensional algebras over an algebraic closed field as well as finite dimensional algebras over a field.

Although my trouble remains,
I would like to know the examples of Noetherian rings which have the infinite Krull dimension.