In order to understand the construction of the quotient ring from last lecture, it is convenient to examine some examples in details. We are interested in ideals of where is a field. We write for the quotient ring, i.e., the set of equivalence classes of polynomials in under the equivalence relation iff
- If then for any the equivalence class is just the singleton and the homomorphism map given by is an isomorphism.
To understand general ideals better the following notions are useful; I restrict to commutative rings with identity although they make sense in other contexts as well:
Definition 1 Let be a commutative ring with identity. An ideal is principal iff it is the ideal generated by an element of i.e., it is the set of all products for
For example, is principal. In every subring is an ideal and is principal, since all subrings of are of the form for some integer