# What are rational integer coefficients?

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?

Sometimes the members of $\mathbb{Z}$ are called "rational integers" to distinguish them from $\mathbb{Z}[i]$ or other rings.
It is common to use the word "integer" to refer not only to elements of $\mathbb{Z}$, but also to a wider class of complex numbers called "algebraic integers": those numbers that are roots of a monic polynomial with integer coefficients.
In this context, "rational integer" is used for elements of $\mathbb{Z}$: it means an integer which is also a rational number.
The word "rational" also gets generalized sometimes. If we have a particular base field of interest, we call elements of that field "rational" and elements of its algebraic extensions "irrational". This reduces to the usual meaning when we take $\mathbb{Q}$ as the base field.
I see this generalization most commonly used when talking about the base field in the abstract, or when working over a particular finite field. e.g. if our base field is $\mathbb{F}_3$, then when discussing the elements of, say, $\mathbb{F}_{27}$, we would call $0,1,2$ rational, and the remaining 24 elements irrational.