Chapter 3 Isomorphisms, Homomorphisms and Ideals

Question 1

What is a homomorphism?

Answer 1

Let R and S be rings
A homomorphism is a map Θ:R->S such that
(i) for all r,r' in R, Θ(r+r')=Θ(r)+Θ(r')
(ii) for all r,r' in R, Θ(rr')=Θ(r)Θ(r')
(iii) Θ(I_R)=I_S

Question 2

What is an isomorphism?

Answer 2

Let R and S be rings
A isomorphism is a bijection Θ:R->S such that
(i) for all r,r’ in R, Θ(r+r’)=Θ(r)+Θ(r’)
(ii) for all r,r’ in R, Θ(rr’)=Θ(r)Θ(r’)

Question 3

What is the kernel of Θ?

Answer 3

Let Θ:R->S be a homomorphism
The kernel of Θ is the set of elements in R that Θ sends to O_S
ker(Θ)={r in R | Θ(r)=O_S}

Question 4

What is an ideal?

Answer 4

An ideal of a ring R is a subset I ⊆ R such that

(i) O within I
(ii) a+b wihin I for all a,b in I
(iii) ar within I and ra within I for all a in I and all r in R

Question 5

What is a principal ideal?

Answer 5

If ainR then {r_1as_1+..+r_nas_n | n>1,r_i,s_i in R} is an ideal which contains a and is the smallest ideal to contain a.It is called the prinicpal ideal generated by a.