# Math Help - prime fields

1. ## prime fields

Let R be a ring with unity 1 and n an integer. Define

n · 1 =

{1 + 1 + . . . + 1 to n terms if n > 0

0 if n = 0

1 + (1) + . . . + (1) to |n| terms if n < 0

(a) Show that ψ : Z R given by

ψ (n) = n · 1

is a ring homomorphism.

(b) Find know rings which are possible images of ψ.

2. (a) Show that ψ : Z →R given by

ψ (n) = n · 1
is a ring homomorphism.
You need to show $\phi (a+b) = \phi(a) + \phi(b)$. And this is true because $(a+b)\cdot 1 = a\cdot 1 + b\cdot 1$. You also need to show $\phi(ab) = \phi(a)\phi(b)$. This is true because $(a\cdot 1)(b\cdot 1) = (1+...+1)(b\cdot 1) = b\cdot 1 + ... + b\cdot 1 = (ab)\cdot 1$. And of course $\phi(1)=1$. Thus, $\phi$ is a (commutative) ring homomorphism.

(b) Find known rings which are possible images of ψ.
$\mathbb{Z}_n$.