# Rings with 0 = 1.

• Nov 25th 2009, 06:12 PM
ENRIQUESTEFANINI
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.
• Nov 25th 2009, 06:24 PM
tonio
Quote:

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
• Nov 25th 2009, 07:42 PM
Shanks
Yes, If 0=1 in R, then R is trivially {0}, containning only the element 0.
• Nov 26th 2009, 12:07 AM
Swlabr
Quote:

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 $a.1= \ldots$
• Nov 26th 2009, 05:34 AM
ENRIQUESTEFANINI
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.