Postby **mobius70** » Thu Jan 31, 2008 6:07 am

Hi Lime

Thanks for your reply.

To answer your question, this is problem from GRE GR9768 Question- 60.

I went through the solutions and all looked kool to me. Well done!!

I was not able to solve for the III) (COMMUTATIVE PART).

You have done it wonderfully.. this is how i solved for first 2 parts.

I ASSUMED THE ADDITION AND MULTIPLICATION ON THE RING TO BE DEFINED AS THE USUAL ARITHMATIC OPERATIONS.

I) Let s be an element of S.

Since S to be a ring S, has to be abelian in addition operation.

=> for every s in S there exists a -s (minus) such that

s+ (-s) = 0 (0 IS THE ADDITIVE IDENTITY)

As defined s= s^2.

=> -s = (-s)^2 = s^2.

=> s = -s.

=> s+s = 0

II) (s)+(t) = (s^2) + (t^2)

also (s+t) = (s+t)^2

=> (s+t)^2 = s^2+t^2

ONLY ONE POINT I WOULD LIKE TO MENTION, FOR AN EXAM LIKE GRE, IT WOULD BE MUCH EASIER TO USE BINARY RING OF ELEMENTS {0,1} TO SOLVE THIS PROBLEM.

I UNDERSTAND THAT THIS APPROACH IS NO WAY SUBSTITUTE FOR A GENERAL SOLUTION AS PRESENTED BY LIME, BUT GUESS IN GRE YOU WANT TO GET ANSWER IN 2 MINS.

Again thanks lime.