Page "Noetherian ring" Paragraph 6

Let denote the ring of integers ; that is, let be the set of integers equipped with its natural operations of addition and multiplication.

An ideal in is a subset, I, of that is closed under subtraction ( i. e., if, ), and closed under " inside-outside multiplication " ( i. e., if r is any integer, not necessarily in I, and i is any element of I, ).

In fact, in the general case of a ring, these two requirements define the notion of an ideal in a ring.

It is a fact that the ring is a principal ideal ring ; that is, for any ideal I in, there exists an integer n in I such that every element of I is a multiple of n. Conversely, the set of all multiples of an arbitrary integer n is necessarily an ideal, and is usually denoted by ( n ).