In my coursebooks and on various websites online (wiki, proofwiki, etc.), among the first theorems which follow the definition of a ring are the uniqueness of the additive identity and the additive inverse. But I haven't found an answer to the following:
Question: If $R$ is a ring with multiplicative identity, is the multiplicative identity unique?
I suspect it is the case, since by definition, $1\cdot r=r\cdot 1=r$ for all $r\in R$; so if we assume that $1$ and $1'$ are two distinct multiplicative identities in $R$, we must have $1=1\cdot 1'=1'$, i.e. $1=1'$ — a contradiction. Is my line of reasoning correct? If so, why is this not included with all the simple theorems?
Best Answer
You are correct, and $1 = 1 \cdot 1' = 1' \Rightarrow 1 = 1'$ is a great proof by contradiction.
As for "not included with all the simple theorems", it has probably already been presented in your book for far "simpler" groups and been treated as already-known facts by this point.