The integers, denoted , are {… -3 -2 -1 0 1 2 3 …}. The natural numbers are the strictly positive integers, denoted . The non-negative integers are denoted . Subsets of are guaranteed to have a least positive element by the well-ordering principle for subsets of .

Below are descriptions of two subsets of . The two cases are explored for structure, then compared for common structure and put in a larger context.

a) Let be a non-empty set of integers such that and for all , . Show there is an such that all elements of are multiples of .

For arbitrary and , one can show elements of form exist by constructing them: , , . Endless similar operations produce the set of elements .

Our objective is to show that all elements in are multiples of some , accomplished by choosing as the least positive integer in . Let’s assume contrarily there is a positive element such that for any integer and derive a contradiction. Consider the sequence of elements of of the form , where ranges over the integers. Pick the value such that is the least positive integer in the sequence. [This equation is called the ‘division algorithm’.] The proof is complete when we show , contradicting that is the smallest positive integer in . Arguing contrarily, if , . Since , it is a member of the sequence, but less than the hypothesized least member , a contradiction. Hence , which in turn contradicts that is the least member of . Hence, all elements in are of form , .

b) Let . Show that any element of is a multiple of .

Let be such that is the least positive integer in . Since contains positive integers, exists. We want to show that and thus and .

Assume , so that and , by the division algorithm. Then . Thus is of the form , in contradiction to the hypothesis that is the smallest such integer. The symmetric argument shows also that .

Let and show . Since and , . Thus . Since is impossible by definition of , we must have . Hence by the abstract argument from part(a), all elements of are multiples of its least positive element .

Case (b) provides a proof that there exist integers such that , by showing that of all such combinations of , the smallest positive one is the .

Abstract algebra provides a structure for describing subsets of as encountered in the cases above. is most generally a commutative ring with identity, meaning that it is both an additive group and a multiplicative group with identity for which .

A ring can have subsets, called ideals, which are additive subgroups of , such that if is an ideal in , then . Every ring has two improper ideals, the set and itself. Some rings, such as , have other proper ideals such as those described herein. In Case (a), candidates for are of course itself, but also the ideals formed by the even integers , or (multiples of 3), etc. To further characterize multiplication in such ideals, and , but .

An ideal of is further described as a principal ideal when each of its elements is an integer multiple of the least positive element of that ideal. We denote principal ideals as where is the least positive element. Thus over , is another label for . Every ideal is by definition a group. Conversely, in , it is easy to show that every additive group is an ideal.

More specifically, is algebraically an integral domain, a commutative ring with distinct additive and multiplicative identity and no zero divisors . At the final level of qualification, is a principal ideal domain (PID), an integral domain in which every ideal is a principal ideal. Every PID is also a UFD as discussed in Problem 5.

Note that set in Case (b) is also an ideal of , the ideal generated by two non-zero integers and via their linear combinations. This concept can be generalized to ideal generators. Let . Define as the set of all finite linear combinations of elements of with integer coefficients. Then it can be shown that is identical to the intersection of all ideals containing . is thus the smallest ideal containing . Define the to be the of all elements of . Then is the smallest positive element of and . This shows a connection between the ideals in Cases (a) and (b), consistent with all ideals of being principal.