Rings and Ideals

In this chapter we introduce some abstract algebra in order to shed some light on several ad-hoc constructions that we have employed previously. In general, a ring is a set on which two compositions called addition and multiplication are defined in such a way that certain axioms hold. In particular, R should be a group with respect to addition and a monoid with respect to multiplication; moreover, distributivity a(b + c) = ab + bc should hold. Here, all our rings will be commutative (ab = ba) domains (ab = 0 implies a = 0 or b = 0; the ring Z/6Z is not a domain because [2][3] = [0]) and will have a multiplicative unit 1 (the ring 2Z of even numbers does not have a unit; sometimes such objects are called rngs). A domain R is called a Euclidean ring if there exists a function ν : R −→ N such that E1 ν(r) = 0 if and only if r = 0; E2 for all a, b ∈ R with b = 0, there exist q, r ∈ R such that a = bq + r and ν(r) < ν(b).