I need help with the following proof:
Show that if :Z Z is a ring isomorphism, then must be the identity mapping. Is there an additive group isomorphism :Z Z other than the identity mapping?
Thanks
and how do you Z that represents the set of integers in Latex.
I have been thinking about this and this is what i came up with...
proof by contradiction
let an isomorphsim that is not the identity mapping.
Then there exists such that
I will assume a is positive(it can be done if a is negative)
we can write a in a few different ways
since Phi is an isomorphism we know that
so know we get
This is a contradiction.
If is a ring isomorphisms then certainly it is a group automorphism of . Group isomorphism on cyclic groups are completely determined on the generator. Since we must have (for that is the definition of a commutative unitary ring homomorphism) it is completely determined, i.e. unique, and in fact the identity mapping.