Generators of ideals

Let’s say we’ve got a ring and an element . What is the smallest left ideal that contains ? Well, we have to have all multiples for so it’s closed under left multiplication. If has a unit, this is all we need. Otherwise, we have to make sure we include all the elements with summands (and their negatives) to make sure it’s an abelian subgroup. Thus the subset is a left ideal in . If has a unit, we just need the subset . We call this the principal left ideal generated by , and write . We can do something similar for right ideals (), and for two-sided ideals we get the subset .

As for any submodules we can form the sum. If we have elements they generate the left ideal , or a similar right ideal. For two-sided ideals we write . The term “principal”, however, is reserved for ideals generated by a single element.

Let’s look at these constructions in the ring of integers. Since it’s commutative, every ideal is two-sided. An integer then generates the principal ideal of all multiples of . In fact, every ideal in is principal.

If is an ideal, consider the subset of all its (strictly) positive elements. Since this is a subset of the natural numbers it has a least element . I say that every element of is a multiple of . If not, then there is some that doesn’t divide. If we can apply Euclid’s algorithm to and , at the first step we get with . The greatest common divisor of and will thus be less than , and Euclid’s algorithm gives us a linear combination for integers and . Thus must be in the ideal as well, contradicting the minimality of .

So every ideal of is principal. When this happens for a ring, we call it a “principal ideal ring”, or a “principal ideal domain” if the ring is also an integral domain.

So how do ideals of integers behave under addition and multiplication? The ideal is the ideal . This it consists of all the linear combinations . In particular, the smallest positive such linear combination is the greatest common divisor of and , as given by Euclid’s algorithm. The product of the ideals is the set of all products of multiples of and : . Thus .