The ring of integers is the set of integers ..., 
,
, 0, 1, 2, ..., which form a ring.
This ring is commonly denoted 
(doublestruck Z),
or sometimes 
(doublestruck I).
More generally, let 
be a number field. Then the ring of integers of 
, denoted 
, is the set of algebraic
integers in 
,
which forms a free Abelian group of rank 
, where 
is the extension degree
of 
over 
.
is also sometimes called the maximal
order of 
.
The Gaussian integers ![Z[i]={a+bi:a,b in Z}](/images/equations/RingofIntegers/Inline15.svg)
form the ring of integers of 
, and the Eisenstein
integers ![Z[omega]={a+bomega:a,b in Z}](/images/equations/RingofIntegers/Inline17.svg)
form the ring of integers of 
, where 
is a primitive
cube root of unity.
