Prove S is a Group
Well, I have another question.
Suppose A is a ring with unity denoted by 1. Let S be the set of elements in A that have multiplicative inverses in A.
I want to prove S is a group under mult. I know the four cases that must be proved. However, I'm having a little difficulty as to how I should prove that these particular things are in the set of elements S.
For example, the last case is to prove that for every element in S there is a multiplicative inverse. However, I am having a hard time thinking as to how I should prove the the inverses reside in S.
If this is vague, I am more than happy to explain what I mean.
Thanks Swlabr....i think i am understanding.
Okay, because this is a ring with unity, it automatically follows that this unity has to be an element of S? (same thing with associative multiplication?) I was thinking along those lines but I kept seeing S as something detached from the ring.
xx' = 1 and yy' = 1. xx' = yy'....? Is this going the write way? Since they are both equal to one is it correct to equate them to each other? however, this is as far as I've gotten. Is closure the same way? Thanks for the help so far.