I have a question about the following excerpt from Atiyah-Macdonald (page 30):
“A ring $A$ is said to be finitely generated if it is finitely generated as a $\mathbb Z$-algebra. This means that there exist finitely many elements $x_1,\dotsc,x_n$ in $A$ such that every element of $A$ can be written as a polynomial in the $x_i$ with rational integer coefficients.”
I suspect one should delete "rational" and then it says that $A$ is called finitely generated if $A = \mathbb Z [a_1, \dots a_n]$ for some $a_i \in A$, that is, every element in $A$ can be written as a polynomial in $a_i$ with integer coefficients.
If this is a typo it is not mentioned on MO but perhaps it is not and I misunderstand the definitions. If I do: What are rational integer coefficients?
Best Answer
Sometimes the members of $\mathbb{Z}$ are called "rational integers" to distinguish them from $\mathbb{Z}[i]$ or other rings.