ABSTRACT
In this chapter we will introduce a particularly nice kind of subring, called a principal ideal, and then generalize yet further to define ideals in an arbitrary commutative ring. It turns out that we will be able to better understand the arithmetic in a given ring by changing our perspective from the properties of individual elements, to considering ideals instead. This was a great insight brought to the subject by such nineteenth century mathematicians as Ernst Kummer and Richard Dedekind: Difficulties in some rings can be better understood by thinking of some ‘ideal’ elements for our ring, in addition to the ordinary elements we have. We will thus begin by translating elementwise properties into statements about corresponding principal ideals, and then discover that non-principal ideals really do fill in certain gaps in our understanding.
