Ring of Integers

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.