Thread: Rings with 0 = 1.

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

Originally Posted by ENRIQUESTEFANINI

Hi:
Let be a ring with unity and assume are the identity elements for addition and multiplication respectively in . Can be true? I think the only way is . 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

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

Originally Posted by ENRIQUESTEFANINI

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

The hint to proving it is

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.