Cosets

Question 1

Cosets

Answer 1

Cosets of an ideal I is the set of I+r where r∈R

Question 2

Add and multiplication of cosets

Answer 2

Add: (I+a)+(I+b)=I+(a+b)
Mul: (I+a)(I+b)=I+ab

This structure ensures that R/I behaves like a ring

Question 3

Well defined operations

Answer 3

The result of the operation must not depend on the specific representative we pick form the cosst

Question 4

Well defined example

Answer 4

If I+a=I+b and I+r=I+s then we need to show
1. I+(a+r)=I+(b+s)
2. I+ar=I+bs

Question 5

Coset theorem

Answer 5

Let I be an ideal of the commutative ring R with a, b ∈ R.

a. If I+a⊆I+b,then I+a=I+b.
b. If I+a∩I+b/=∅,then I+a=I+b.
c. I+a=I+b if and only if a−b∈I.
d. There exists a bijection between any two cosets I + a and I + b. Thus, if I has finitely many elements, every coset has that same number of elements.

Question 6

The Natural Homomorphism

Answer 6

Suppose that R is a commutative ring with ideal I. Consider the function ν: R → R/I defined by

ν(a) = I + a.

ν is a homomorphism from R onto R/I. We call it the natural homomorphism from R onto R/I.