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Landau
#4
Feb8-11, 03:03 PM
Sci Advisor
P: 905
1=0 only in trivial {0} ring.

Quote Quote by tc_11 View Post
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.