# 1=0 only in trivial {0} ring.

tc_11
Hi, I found a couple of proofs proving that 1=0 only in the trivial ring {0}. They say
Suppose 1 = 0. Let a be any element in R; then a = a ⋅ 1 = a ⋅ 0 = 0.

But what I don't understand is that they say a = a ⋅ 1. But that is only true if a ring has unity (x*1=1*x=x), and it is possible to have a ring without unity, so why is it okay to say a = a ⋅ 1?

What could 1 possibly mean in a ring without unity? tc_11
Yeahh okay that's what I was thinking. 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. Thanks!

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.

tc_11
Okay.. but if we are talking about a ring where 1=0, don't we already know 1 is in the ring?

Okay.. but if we are talking about a ring where 1=0, don't we already know 1 is in the ring?
I don't know how I can be more clear than in my last post:
So once you state a property about "1" you are assuming such an element exists in the first place.

tc_11
I'm sorry, I'm just trying to understand... my initial question is: we want to prove that the only time 1=0 is in the trivial ring {0}. And in the proof, it is said a=a*1. And so I am trying to clarify... we can use the property a=a*1, because we are talking about a ring where 1=0, we know the ring contains the identity element 1 since 1=0 in our ring? http://en.wikipedia.org/wiki/Proofs_of_elementary_ring_properties

we can use the property a=a*1, because we are talking about a ring where 1=0, we know the ring contains the identity element 1 since 1=0 in our ring?
I am trying to explain that the statement "1=0 holds in our ring" does not have any meaning, except if it had already been assumed that our ring contains the identity element 1. You can't prove a meaningless statement. If our ring does not contain 1, what do you think it would mean to say 1=0?

tc_11
So if our ring does not contain 1... then our ring does not have unity (there is no element such that a*1=a). Then 1=0 would mean... I'm not sure.. that the only element must be 0 because 1's not in there?

espen180
I think this sounds like a contradiction. First you say 1 is not in R. Then you say 1=0 leads to R={0}?

What do you think is confusing about Landau's statement?

Let R be a ring with 1. If 1=0, then R={0}.