MATH21112 Rings and Fields
Example Sheet 6
More on Homomorphisms and Ideals
1. Show that the map from Z[√2] to itself given by sending a + b√2 to
a -b√2 is an automorphism. Show that there are no other automorphisms of Z[√2] apart from the identity map.
2. Let R = Z [√2] and let Define the map Prove that θ is an isomorphism.
(You may assume that S is a ring.)
3. Prove that if θ : R -→ S is a surjective homomorphism of rings and R is commutative then S must be commutative. Give an example to show that the conclusion may be false if θ is not surjective.
4. Suppose that R is a commutative ring of characteristic 3. Prove that the map θ : R -→ R defined by θ(r) = r3 is a homomorphism. What is ker(θ)?
5. Give an example of a ring R and a subset H of R which contains 0 and is closed under addition and multiplication but which is not an ideal of R.
6. Prove that if θ : R -→ S is a surjective homomorphism of rings and I is an ideal of R then θ(I) is an ideal of S. Give an example to show that the conclusion may be false if θ is not surjective.