Quote by tc_11
We know 1 is in R.... and there is no other way for the number one to behave... 1*x = x always. And so since 1 is in R, we must have unity.

This is not really formulated correctly. The element "1", pronounced "the identity element" or "unit element" is
by definition an element with the property that 1x=x=x1 for all x. So once you state a property about "1" you are assuming such an element exists in the first place.
So the correct statement should be:
Let R be a ring with 1. If 1=0, then R={0}.
The first sentence is essential, because otherwise the second sentence does not make any sense.