Ring Homomorphisms, Ideals, Quotient Rings, and the First Isomorphism Theorem
Today we shall arrive at a very important result about rings then use it to show a rather interesting result involving the complex numbers!
(Here is a link to the post I made about the definition of a ring)
Ring Homomorphisms:
Broadly, a homomorphism is a map between structures of the same type which preserves that structure. For example, linear maps are vector space homomorphisms because they preserve vector addition and scalar multiplication. In our case, ring homomorphisms preserve the addition and multiplication structure of rings. More formally:
Definition: Let R and S be ring and φ:R->S a map between them. Then we say φ is a homomorphism if ∀x,y∈R we have
Note: here I have used the subscripts to emphasise when the operation the one acting on R or on S and to distinguish between the multiplicative identities of R and S in the first line. From here on, I won't use this convention as context usually makes this clear.
Example:
We will use this example later to prove a result involving the complex numbers!
From the definition of a homomorphism we easily get two nice results!
Lemma:
We can also consider a special type of homomorphisms called isomorphisms. Intuitively, an isomorphism is a map between two rings whose operations behave the same way, i.e. the only real difference between the rings is how we label elements. More formally:
Definition:
Example:
Now I wil define two sets that play an important role in what follows!
Definition:
Note: The kernel is a subset of R and the image is a subset of S. In fact, the image is a subring of S!
Lemma:
Here is a page about the Subring Criterion
Ideals:
We've seen that the image is a subring of S but what about the kernel? It can't be a subring of R because it doesn't contain the multiplicative identity by definition of homomorphism (unless the multiplicative and addative identities are the same, but then we have the trivial ring). However we do have some nice properties!
Lemma:
This leads us to the following definition:
Definition:
Remark: We can define separately left and right ideals where left ideals only need contain rx and right ideals only need contain xr. For commutative rings rx=xr so the two notions are equivalent and are equivalent to our definition of an ideal. But for non-commutative rings that isn't necessarily true which is why we include both sides in our definition.
Example:
Note: since R is commutative here we didn't need to check right multiplication since rax=rxa.
Cosets and Quotient Groups:
The aim here is to construct new rings from rings we already know about and in the process generalise the notion of modulo arithmetic. A key observation here is that for the integers there is a link between residue classes modulo n and the principal ideal generated by n:
This leads to the following definition:
Note: The ideal is equal to the trivial coset: I=0+I.
We now see that cosets partition R in the following lemma!
Lemma:
This means that we can have multiple representations of the same coset since for any x'∈x+I, the lemma says x'+I=x+I.
With this, we can now define what a quotient ring is!
Definition:
We need to check that this is well-defined, that is we need to check that these operations don't depend on our choice of respresentation of our cosest:
Then the addative identity of R/I is I itself and 1+I is the multiplicative identity. The fact that R/I is an addative group follows easily from the fact that R is an addative group and similarly the properties of multiplication (i.e. associativity and distributivity) follow easily from the corresponding properties of R.
We then called R/I the quotient ring of R by I or "R mod I" and we have a homomorphism φ:R->R/I given by φ(r)=r+I. This is called the canonical map and the homomorphism properties follow directly from the definitions of addition and multiplication on R/I. The kernel of the canocial map is always the ideal I which shows that all ideals are in fact that kernel of some homomorphism!
The First Isomorphism Theorem:
We can actually say more about quotient rings! The following theorem is extremely important in the study of rings!
Theorem:
Now it is time for one final example involving the an interesting way to think about the complex numbers!
Example:
Note: In particular, the element that behaves like i in ℝ[x]/(x^2+1) is x+(x^2+1). This is seen quite easily from the evaluation map since φ(x)=i but it can also be seen from the fact that ker(φ)=(x^2+1)!
