Elementary Abstract Algebra- Examples and Applications, 2019a

856 CHAPTER 20 INTRODUCTION TO RINGS AND FIELDS
(a) Show that φ is an isomorphism.
(b) Give another isomorphism between Q[x, y] andQ[x, y].
When we restrict the isomorphism, so that R 1 = R 2 , we have a special
type of isomorphism known as an automorphism.
♦
Definition 20.5.11. A ring automorphism is a ring isomorphism whose
domain is equal to its range.
△
Example 20.5.12. Show that f(a + bi) =a − bi is a ring automorphism
from C to C.
We showed in Example 20.5.8 that this function is a ring isomorphism.
It should be clear that the domain and range of f are the same, so f is also
a ring automorphism.
Exercise 20.5.13.
a, b, c ∈ R.
Consider the function f((a, b, c)) = (a, −b, c), where
(a) show that f is a homomorphism by proving that:
(1) f((a, b, c)+(d, e, f)) = f((a, b, c)) + f((d, e, f)), and
(2) f((a, b, c) · (d, e, f)) = f((a, b, c)) · f((d, e, f)).
(b) Is f an isomorphism? (*Hint*)
(c) Is f an automorphism?
♦
20.6 Ring homomorphisms: kernels, and ideals
As we have seen above, ring isomorphisms are functions that are bijections
and preserve the additive and multiplicative operations. It is possible to
have functions that are not bijections but still preserve the additive and