TITLE: What's the difference between the ring of Integers and the maximal order in a Number field?
QUESTION [6 upvotes]: I have faced the concept of maximal order, while reading one of the books of Goro Shimura, Euler Products and Eisenstein Series.
I am familiar with the concept of ring of integers of a number field. I have read something on the orders and lattices, but its confused me.
So at first, I prefer to fix some definitions to reduce my confusion.
Let $\mathbf{F}$ be a number field, i.e. a finite extension of $\mathbb{Q}$. Let $n$ to be the degree of this extension, i.e. $n:=[\mathbf{F}:\mathbb{Q}]$, also we denote by $\mathfrak{O}_{\mathbf{F}}$ the ring of integers of $\mathbf{F}$.
By an order $\mathfrak{O}$ in $\mathbf{F}$ we mean a subring of $\mathfrak{O}_{\mathbf{F}}$, which has the $\mathbb{Z}$-dimension equal to $n$.
Are all of my definitions are right, Specially the definition of an order?

REPLY [9 votes]: Yes, that is the definition of an order in a number field. And indeed
in a number field, the ring of integers is the unique maximal order.