Re: Please help, Thanks!

On Mon, 17 Mar 2008 18:40:58 EDT, steven <chasedtime@xxxxxxxxx> wrote:

define a ring R={0,1,a,b} by taking table below as its multiplication table.
* | 0 1 a b
-------
0 | 0 0 0 0
1 | 0 1 a b
a | 0 a b 1
b | 0 b 1 a

To define a ring, you also have to define how elements _add_.

So you only have half a ring (just kidding).

prove that as rings, R is not Isomorphic to Z_2xZ_2.

Hint: Count the number of elements of order 2 (elements x such that
x^2 =1).

Alternate hint: Note that R is commutative, and that all nonzero
elements have inverses -- hence R is a field. But, assuming you have,
as a known theorem, the fact that the multiplicative group of a finite
field is cyclic, it follows that a,b have order 3. Now look at Z_2 x
Z_2.

quasi
.

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