Which Elements are the 0 and 1 in this Commutative Ring with Unit?

[fk378]

Homework Statement

Given a set R={x,y,a,b}

There are 2 tables shown: one is the addition of x,y,a,b elements with x,y,a,b elements. The 2nd table is multiplication of each element with the other elements. (The 2nd table shows that x multiplied by anything equals x)

we have a+b=y and (a)(b)=y

Decide which elements must be the 0 and the 1, then prove that this is a commutative ring with unit.

The Attempt at a Solution

I know how to show it is commutative. I'm just having trouble starting on it.

Well if the addition of a and b equals the multiplication of a and b, then a=b=y=0. Is this right? But then since x multiplied by anything equals x, then x must also be 0. But then there is no element 1. So how can I show commutativity with this??

Also don't know how to show that there exists units.

 

[tiny-tim]

Hi fk378!

It would help if you showed us the table …

either use the code tag like this…

a b c d
e f g h
i j k l
m n o p

or use LaTeX and http://www.physics.udel.edu/~dubois/lshort2e/node56.html#SECTION00850000000000000000