After Midterm

Question 1

Definitions Ring Homomorphism

Answer 1

A ring homomorphism φ from a ring R to a ring S is a mapping from Rto S that preserves the two ring operations; that is, for all a, b in R
φ(a+b)=φ(a)+φ(b)
φ(ab)=φ(a)φ(b)
φ(1R)=1S

Question 2

A homomorphism φ from R[x]===>A is the same data as ψ: R==>A together with an element a ∈A (a=φ(x))

Answer 2

A homomorphism φ from R[x]===>A is the same data as ψ: R==>A together with an element a ∈A (a=φ(x))

Question 3

If φ is homomorphism then φ(Or)=Os

Answer 3

Proof

φ(Or)= φ(Or+Or)= φ(Or)+s φ(Or)
so Os+s φ(Or)=φ(Or)+s φ(Or)
==>Os=φ(Or) since (S,S+)

Question 4

Isomorphism

Answer 4

φ: R==>S if is a bijective Homomorphism