Introduction to Rings

Abstract

In the previous chapter we have seen that the integers possess a division algorithm and that from this division algorithm there may be derived a Euclidean Algorithm for finding the greatest common divisor of two given integers. ‘Polynomials’ share many properties in common with the integers, having a division algorithm and a corresponding Euclidean Algorithm. As our treatment of polynomials proceeds, initially somewhat informally, it will become apparent that we need to consider much more precisely the extent to which integers and polynomials share common features. In this way we shall be led to enunciate axioms for an algebraic system called a ‘ring’ and for a ring of a particular type called an ‘integral domain’ which incorporates some of the features common to integers and polynomials.