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
Definition 2 An integral domain is a commutative ring with identity wher and without zero divisors. Recall that is a zero divisor iff and there is some such that
For example, in any nonzero with is a zero divisor: Let and consider Then
Definition 3 A unit in a commutative ring with identity is an for which there is some such that We write for the set of units of
For example, in any with is a unit: Recall that if then there are integers such that so if then
If is a field, since every nonzero element has an inverse.
If is a field, then To see this, recall that the degree of a polynomial is if and it is if the largest power of that appears in with a nonzero coefficient of is Suppose that are nonzero polynomials in Say that and so there are coeffiecients and with not zero, such that
Then since the coefficient of in the product is the sum of all the coefficients with Of course, this is nonzero only if and so Moreover, the coefficient of in is which is nonzero since a field has no zero divisors.
This shows that if both are nonzero. If one of them is zero, then as well, and is true due to our convention that
It is clear now that if is a unit in then since otherwise for any nonzero but then But means that is a nonzero constant, i.e., a nonzero element of
Notice that this argument also shows that is an integral domain, as we showed that whenever both are nonzero.
Definition 4 An integral domain is a principal ideal domain (pid) iff every ideal of is principal.
For example, any field is a principal ideal domain, although this example is somewhat trivial:
More generally, suppose is a commutative ring with identity and let be an ideal of Suppose that contains a unit. Then because if is a unit then there is some such that and so for any Since ideals are closed under multiplication by any element of it follows that Since was arbitrary,
Now, if is a field, then every nonzero element of is a unit, and we have that any ideal is either or
A more interesting example if That this is a pid is shown in detail in Chapter 8 of the book, that I highly recommend you study carefully. The argument resembles strongly results we shoed for the integers.
The idea is this:
- First, in we have a division algorithm,
so given any nonzero polynomials in there are unique polynomials in with and such that Note in particular that if i.e., if is a (nonzero) constant, then as must be zero.
In fact, we can generalize to the notion of greatest common divisor:
Definition 5 If are nonzero polynomials in a greatest common divisor (gcd) of is any nonzero polynomial such that:
- Whenever is a polynomial in and and then
So any nonzero polynomials may admit more than one gcd, since if is a gcd of then so is for any On the other hand, this is the only obstacle to uniqueness: If are both gcds of then and so and there is a unit such that
- Second, in we have an Euclidean algorithm,
so by repeated application of the division algorithm we can find a gcd of any two nonzero polynomials moreover, there are polynomials such that is a gcd of
Using this one can easily show that any ideal in is principal, just as we did for We’ll revisit the details after the break, and they are in Chapter 8 of the book, but it is important that you notice that we are basically repeating the proofs we already know.
Just as with once we have a notion of gcd and an Euclidean algorithm, we can talk about prime or irreducible elements in Once we have this notion, we will revisit the construction of the quotient rings and study their properties.
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