Section 5 - Ring Homomorphisms

Question 1

Ring Homomorphism

Answer 1

Let R and S be rings. A ring homomorphism is a map: φ: R → S such that the following hold for all a,b∈R:
(i) φ(a + b) = φ(a) + φ(b). (φ respects addition)
(ii) φ(ab) = φ(a)φ(b). (φ respects multiplication)

Question 2

Given φ: R → S, φ(0R) = ???

Answer 2

φ(0R) = φ(0S)

Question 3

If φ: R → S, what is Ker(φ)?

Answer 3

Ker(φ) = {a∈R | φ(a) = 0S}

Question 4

How can you determine if a ring homomorphism is injective?

Answer 4

A ring homomorphism φ is injective iff Ker(φ) = {0R}

Question 5

Image of a Ring Homomorphism

Answer 5

If φ: R → S is a ring homomorphism, then Image(φ) is a subring of S

Question 6

Isomorphism

Answer 6

A ring homomorphism φ: R → S is called an isomorphism if there is a ring homomorphism Ψ: S → R such that: Ψφ = 1R and φΨ = 1S (i.e. Ψ(φ(r)) = r, ∀r∈R and φ(Ψ(s)) = s, ∀s∈S).

Question 7

Inverse of Ring Homomorphism

Answer 7

If φ is an isomorphism, the ring homomorphism Ψ (where Ψφ = 1R and φΨ = 1S) is unique, and we call Ψ the inverse of φ (written φ^-1)

Question 8

What are the differences between two rings that are isomorphic?

Answer 8

Isomorphic rings are algebraically the same rings. Only the symbols representing their elements are different

Question 9

What statements are equivalent to φ being an isomorphism?

Answer 9

Let φ: R → S be a ring homomorphism. Then the following are equivalent:
(i) φ is an isomorphism
(ii) φ is bijective (i.e. injective and surjective)
(iii) Ker(φ) = {0R} and Image(φ) = S

Question 10

First Isomorphism Theorem

Answer 10

Let φ: R → S be a ring homomorphism. Then there is a well-defined ring isomorphism:
φ: R/Ker(φ) → Image(φ) with the mapping: a + Ker(φ) → φ(a)