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 is a ring of dimension over , where is the extension degree of over . is also sometimes called the maximal order of .
The Gaussian integers is the ring of integers of , and the Eisenstein integers is the ring of integers of , where is a primitive cube root of unity.