Remark 1. Consider an arbitrary ring (R,+, •) and a nonempty subset S of R. Then the set S can be defined by
(S)=?{I: S ? I, (I,+, •) is an ideal of (R,+, •)}.
This set is a collection of all ideal which contain S which is not empty, since the improper ideal belong to it.
Theorem 1. The triple ((S),+, •) is an ideal of the ring (R,+, •), known as the ideal generated by the set S.
Let (I,+, •) be any ideal of (R,+, •) for which S ? I, then (S) ? I. It can be considered ((S),+, •) as the smallest ideal contains the set S.
Definition 1. An ideal generated by a single ring element, say a, is called a principal ideal and is denoted by ((a),+, •).
Theorem 2. If (R,+, •) is a commutative ring with identity and a ? R, then the principal ideal ((a),+, •) generated by a such that
(a) = {r • a : r ? R}.
The principal ideals are the only ideals of the ring of integers, as shown in the following theorem.
Theorem 3. If (I,+, •) is an ideal of the ring (Z,+, •) then I = (n) for some nonnegative integer n.
Proof: If I = {0}, the theorem is trivially true, since ({0},+, •) is the zero ideal and it is principal ideal generated by 0.
Suppose I contain other elements in addition to zero. Let +m; - m ? I and n is the least positive integer in I. As (I,+, •) is an ideal, each integral multiple of n must be in I. so
(n) ? I (1)
On the other hand, any integer k ? I can be written as k = qn+r (from division algorithm), where q; r ? Z and 0 ? r < n. Since k and nq ? I so k - qn = r ? I. From definition of the integer n then r = 0, hence
k = qn. Thus, for all element in I is a multiple of n. This means that
I ? (n) (2)
From Equations (1) and (2), I = (n).
Remark 2.
1. A principal ideal ring is a commutative ring with identity such that every ideal is principal.
2. From Thm ( ), the ring of integers forms a principal ideal ring.
Remark 3. Suppose that a1, a2,..., an are nonzero elements of (R,+, •) with (R,+, •) is a commutative ring with identity. An element a ? R is said to be a common multiple of a1, a2,..., an provided a is a ring multiple of each of these. For example, the products a1 • a2... an and -( a1 • a2... an) are both common multiples of a1, a2,..., an.
We shall call the element a is least common multiple (LCM) of a1, a2,..., an if
1. a is a common multiple of these elements and
2. any other common multiple of a1, a2,..., an is a multiple of a as well.