In ring theory, an ideal is defined as a subset of a ring with the following properties:
- Zero is an element of the ideal
- The ideal is closed under addition
- The product of an element of the ideal and an element of the initial ring is an element of the ideal
In general, one must distinguish between left and right ideals, because many rings do not have commutative multiplication. All discussion here applies to commutative rings.
An example is the set of even integers, as a subset of the ring of integers. This is an ideal because:
- 0 is even.
- The sum of two even integers is even.
- The product of an even integer with and any other integer is even.
This example is itself an example of a principal ideal. Given any ring R and element , we may define an ideal (r), which consists of all elements of R which may be written as the product of r with some other element a of the ring. The ideal (r) is called a principal ideal. It is an ideal, because:
- If and , then we can write x = ra and y = rb. Then .
- If and is an arbitrary element of the ring R, then .
When is the ring of integers, then every ideal is a principal ideal: that is, any ideal I of the ring of integers is the set of elements "multiples of k" for some integer k. Many familiar rings have the property that every ideal is a principal ideal: such rings are known as principal ideal domains.
In Polynomial Rings
One familiar ring is , the ring of polynomials over the integers. One ideal in R is the polynomials with constant term 0, such as x + 5x3. This can be thought of as the set of polynomials p satisfying p(0) = 0. This is an ideal, because:
- The zero polynomial has constant term 0.
- If f,g are two polynomials with constant term 0, then (f + g)(0) = f(0) + g(0) = 0 + 0 = 0.
- If f has constant term 0 and p is arbitrary, then .
A brief reflection shows that this ideal is actual just the principal ideal (x): every polynomial vanishing at 0 is just the polynomial x times some other polynomial. In fact, it turns out that is a principal ideal domain.
However, consider the ring of polynomials in two variables. There is again an ideal I of polynomials vanishing at 0. However, this is not a principal ideal. If it were principal and generated by p(x,y), then this p would divide both the polynomials x and y, since both are in the ideal. Clearly there is no such polynomial. This I is an exactly of an ideal that is not principal, and is a ring which is not a principal ideal domain.
Prime ideals provide a way to generalize the prime numbers. An ideal I is called a prime ideal if implies that either or . The basic examples of prime ideas are the principal ideals , where p is a prime number. For example, consider the ring (5), which consists of all integer multiples of 5. If the product of two numbers is a multiple of 5, it must be that one of the numbers is a multiple of 5. This is in contrast with ideals that are not generated by a prime number: for example, (6) is the ideal of all multiples of 6. However, it is possible for the product of two numbers to be a multiple of 6 without either of the numbers being one: for example, .