Ring and Groups

**rajr** · March 17th 2009, 08:53 AM

This was a challange question and I am not sure how to do this.. help would be appreciated thx

**stapel** · March 17th 2009, 09:39 AM

$\mbox{a) }\, (a\, +\, b)^2\, =\, (a\, +\, b)(a\, +\, b)\, =\, (a\, +\, b)a\, +\, (a\, +\, b)b$

$=\, a^2\, +\, ba\, +\, ab\, + \, b^2\, =\, a\, +\, ba\, +\, ab\, +\, b\, =\, (a\, +\, b)$

Subtract $(a\, +\, b)$ from either side. See if that leads anywhere helpful...?

$\mbox{b) }\, (ab)^2\, =\, (ab)(ab)\, =\, abab\, =\, a^2 b^2$

$a^{-1}ababb^{-1}\, =\, a^{-1}a^2 b^2 b^{-1}$

$ebae\, =\, eabe$

...where $e$ is the group's identity element. The result follows immediately.

**rajr** · March 17th 2009, 09:48 AM

Originally Posted by stapel

$\mbox{a) }\, (a\, +\, b)^2\, =\, (a\, +\, b)(a\, +\, b)\, =\, (a\, +\, b)a\, +\, (a\, +\, b)b$

$=\, a^2\, +\, ba\, +\, ab\, + \, b^2\, =\, a\, +\, ba\, +\, ab\, +\, b\, =\, (a\, +\, b)$

Subtract $(a\, +\, b)$ from either side. See if that leads anywhere helpful...?

$\mbox{b) }\, (ab)^2\, =\, (ab)(ab)\, =\, abab\, =\, a^2 b^2$

$a^{-1}ababb^{-1}\, =\, a^{-1}a^2 b^2 b^{-1}$

$ebae\, =\, eabe$

...where $e$ is the group's identity element. The result follows immediately.

thx for the reply and I can understand the second part and for the first part if I takeaway a+b from both sides then I will get ab = -ba but is it ok to say that it is commutative just from that step or do I have to go further any more steps to prove it

I think for the part (ii) I have to do like as follows:

b(a+a) = 0;
therefore a+a =0 is it right
thx once again for ur help

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