# Thread: Is this a ring homomorphism?

1. ## Is this a ring homomorphism?

Hi, could you please help with the following question. I need to understand the method in proving it's a homomorphism and then i can do other similar questions i've been asked to complete.

Let Z x Z be the ring which is the cartesian product of two copies of Z. Determine if the following is a ring homomorphism.

ZxZ -> Z, h(x,y)=x

(Z is the set of integers, and -> is an arrow)

Thanks for you help.

2. Originally Posted by Louise
Hi, could you please help with the following question. I need to understand the method in proving it's a homomorphism and then i can do other similar questions i've been asked to complete.

Let Z x Z be the ring which is the cartesian product of two copies of Z. Determine if the following is a ring homomorphism.

ZxZ -> Z, h(x,y)=x

(Z is the set of integers, and -> is an arrow)

Thanks for you help.
Remember to be a homomorphism it must preserve addition and multiplication.

let $\displaystyle (a,b) \in \mathbb{Z} \times \mathbb{Z}$ and $\displaystyle (c,d) \in \mathbb{Z} \times \mathbb{Z}$

$\displaystyle h[(a,b)+(c,d)]=h[(a+c,b+d)]=a+c=h[(a,b)]+h[(c,d)]$

Since you didn't define mult on ordered pairs I'm not sure about your ring ops, But it would be verified in the same manner.

I hope this helps.

3. ## To show: the set of complex numbers of the form a+bi where a,b in Q is isomorphic to

Thanks so much for your help.

I have another question on rings and fields;

Let F be the set of all complex numbers of the form a+bi with a, b in Q. (Q= rationals) Show that F is a subring of C (C=complex numbers), and that it is a field. Show that F is isomprphic to the field of fractions of the ring Z[i] of Gaussian integers.

I have shown that F is a subring of the complex numbers using the subring test.

Is it correct to prove F is a field by showing it has a one (1), is not the zero ring, every non-zero number has an inverse (a/(a^2 +b^2) -(b/(a^2 +b^2))i) and that it is commutative (a+bi) +(c+di) = (c+di) + (a+bi)?

Finally is showing that F is isomorphic to the field of fractions of the ring Z[i] of Gaussian integers the same as showing that the Guassian numbers are isomorphic to the field of Gaussian rationals.

How do i show that these are isomorphic?

Thanks a lot for your help.