Expert Answer to "If the cardinality of a ring R is..."

Dereon Guzman

Answered question

2022-04-21

If the cardinality of a ring R is greater than 1, then

1 \neq 0

bobthemightyafm

Beginner2022-04-22Added 16 answers

Step 1
Well, suppose R is a ring with the property that

0 = 1

and

| R | \geq 2;

hence there is an element

x \in \frac{R}{{i}}

where i is the elements where

i = 0 = 1

Then

i x = 0

since

i = 0

i x = x

since

i = 1

but then

x = 0

, a contradiction. Hence,

0 \neq 1

in R
The lone case where

0 = 1

thus only happens in the trivial ring

R = {0}

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?