# Thread: Rings with 0 = 1.

1. ## Rings with 0 = 1.

Hi:
Let $R$ be a ring with unity and assume $0,1$ are the identity elements for addition and multiplication respectively in $R$. Can $0=1$ be true? I think the only way is $R=(0)$. Any hint will be welcome. Regards.

2. Originally Posted by ENRIQUESTEFANINI
Hi:
Let $R$ be a ring with unity and assume $0,1$ are the identity elements for addition and multiplication respectively in $R$. Can $0=1$ be true? I think the only way is $R=(0)$. Any hint will be welcome. Regards.

Indeed, and that is why many authors disqualify 0 = 1 or explicitly require that these are two different elements when dealing with rings.

Tonio

3. Yes, If 0=1 in R, then R is trivially {0}, containning only the element 0.

4. Originally Posted by ENRIQUESTEFANINI
Hi:
Let $R$ be a ring with unity and assume $0,1$ are the identity elements for addition and multiplication respectively in $R$. Can $0=1$ be true? I think the only way is $R=(0)$. Any hint will be welcome. Regards.
The hint to proving it is $a.1= \ldots$

5. Thanks, thanks. I'd seen definitions requiring the ring to have two or more elements, but I'd never seen the same thing specified as zero and one being diferent elements. This seemed a bit odd to me. Now I see that authors use these two constrains interchangably. Thanks again,

Enrique.