1. ## One Sided Identities

I'm having trouble with this problem. Here is all the problem states:
Prove: Let <A, O> be a system (where O is the operator) with a left identity of g and a right identity of d. (So for all elements of A, a, aOd = a and gOa = a)
Show that g = d.

As corollaries, show that a two-sided identity is unique, and if O is commutative, then <A,O> has at most one left - identity.

I understand what they are asking for but I can't figure out how to prove it without assuming O is commutative. Thanks for the help in advance.

2. Originally Posted by jameselmore91
I'm having trouble with this problem. Here is all the problem states:
Prove: Let <A, O> be a system (where O is the operator) with a left identity of g and a right identity of d. (So for all elements of A, a, aOd = a and gOa = a)
Show that g = d.

As corollaries, show that a two-sided identity is unique, and if O is commutative, then <A,O> has at most one left - identity.

I understand what they are asking for but I can't figure out how to prove it without assuming O is commutative. Thanks for the help in advance.
$\displaystyle gOa=a\;\forall\;a\in A$ $\displaystyle \implies$ $\displaystyle gOd=d$.

$\displaystyle aOd=a\;\forall\;a\in A$ $\displaystyle \implies$ $\displaystyle gOd=g$.

The conclusion follows.