Relations and Maximal Ideals

Fact Imposing Relations We interpret the quotient as imposing the relations or ‘killing’ the elements . Killing the elements in installments and in various orders, or in one strike lead to the same result. That is, we have following commutative diagram:

Adjoining Elements

A primary tool for constructing new rings is to take a ring and a ‘free’ variable and create the polynomial ring, and then quotient by a polynomial. We show see this further in field extensions by adjoining roots of polynomials.

e.g. The complex numbers are constructed from by setting . In fact, we can define . Clearly it is surjective and injective. Since $$ \begin{align} \varphi(a+bi)\varphi(c+di)&=\overline{ac+(ad+bc)x+bdx^2} \ &=\overline{(ad+bc)x-bd+ac} \ &=\varphi((ad+bc)i+(ac-bd)) \ &=\varphi((a+bi)(c+di)), \end{align} $$$\varphi$ is homomorphism, yielding a isomorphism.

Proposition

If in has no root in we can form the quotient and is a root of .

Definition

Whenever we have a subring of a ring and an element we denote by the smallest subring of that contains and .

Proposition

defines an isomorphism .

Field of Fractions & Product Rings

The Field of Fractions

Suppose is an integral domain. Consider equip with the following relation: This is an equivalence relation. Let denote the quotient. We denote the equivalence classes by .

Proposition

is a field.

Theorem

Suppose is an integral domain and is defined as above. Define addition and multiplication on byThen is a field, denotes as . The map given by is an injective ring homomorphism.

e.g. If then . If , for a field , then the field of fraction of , denoted , is called the field of rational functions over :

Def Product Rings Let be rings. Then , with coordinate-wise addition and multiplication, is a ring with zero and identity element .

Prime and Maximal Ideals

Prime Ideal

Let be a ring. An ideal is prime if and implies that either or .

Theorem

Let be a ring. is prime if and only if is an integral domain.

Maximal Ideal

Let be a ring. An ideal is maximal if and there is no other ideal such that .

e.g. is maximal iff for a prime . In general, we have the following proposition holds:

Proposition

Let be a principle ideal domain, then is maximal if and only if is irreducible.

Proof In a PID, we have that iff . It follows that is maximal iff there is no proper divisor , which means is irreducible.

Corollary

If is a field, then is maximal iff is irreducible.

Proof is a PID, so is maximal iff is irreducible.

Theorem

Let be an ideal. Then is maximal iff is a field.

Corollary

Every maximal ideal is prime.

Proof Every field is an integral domain.

Corollary Let be an irreducible polynomial. Then is a field and contains a root of , namely the coset .