The definition "module of finite type".

Question

I know the definition of "module of finite type"

Is that different from finitely generated module ?

Thanks.

Answer 1

This term is sometimes used to mean a finitely generated module. That is to say, if someone says ``an $R$-module of finite type," he or she definitely means finitely generated as a module. I think that nowadays it is more common to say "finitely generated $R$-module," or, especially in commutative algebra, "finite $R$-module." I prefer this terminology. Of course the latter could lead to confusion in principle, but not usually in practice. In the contexts of $R$-algebras, an $R$-algebra of finite type is definitely not the same thing as an $R$-algebra that is finitely generated as an $R$-module (often just called a finite $R$-algebra).

Answer 2

finite type means finitely generated $R$- algebra
finite means finitely generated $R$-module
so in Atiyah-Macdonald's famous book this proposition

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