I know the definition of "module of finite type"

Is that different from finitely generated module ?



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).


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


enter image description here

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    $\begingroup$ You're talking about algebras, the OP is talking about modules. $\endgroup$ – Najib Idrissi Apr 17 '14 at 19:40
  • $\begingroup$ im talking about difference of finite type and finite $\endgroup$ – user142666 Apr 18 '14 at 11:09

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