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?

Solution 1:

Sometimes the members of $\mathbb{Z}$ are called "rational integers" to distinguish them from $\mathbb{Z}[i]$ or other rings.

Solution 2:

In algebra there are lots of different situations where "integers" arise, as in the context of algebraic integers. The term rational integer just distinguishes the normal integers from any other sort of algebraic integer.