Prove that the field R of real numbers is isomorphic to the ring of all 2x2 matrics of the form (0 0; 0 a) with a a real number.

It says to let f(a)=(0 0; 0 a), but I don't know how to show that this is one to one and onto.

Thanks.

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- Feb 18th 2010, 07:09 AMzhupolongjoeDefining an isomorphism....
Prove that the field R of real numbers is isomorphic to the ring of all 2x2 matrics of the form (0 0; 0 a) with a a real number.

It says to let f(a)=(0 0; 0 a), but I don't know how to show that this is one to one and onto.

Thanks. - Feb 18th 2010, 07:15 AMSwlabr
Are you sure you don't mean matrices of the form

$\displaystyle \left( \begin{array}{cc}

a & 0 \\

0 & a \end{array}

\right)$?

As the set you gave has no identity.

Think about it for a bit - what does surjective mean? take an arbitrary matrix of this form. Is this mapped onto by something? What if two matrices were mapped to by the same element, what would this mean with respect to the elements which were mapping to it?

Are you confident with the definitons of surjectivity and injectivity? - Feb 18th 2010, 07:18 AMzhupolongjoe
No, the book says matrices of the form (0 0; 0 a). Doesn't this have identity

(0 0; 0 1)? Identity doesn't have to be (1 0; 0 1) - Feb 18th 2010, 07:19 AMSwlabr
- Feb 18th 2010, 07:22 AMclic-clac
When are two matrices equal? I guess answering this question will instantaneously make your map become one-to-one and onto.

- Feb 18th 2010, 07:48 AMzhupolongjoe
Okay, I think I got it actually. Maybe I was just making this problem harder than it really is.