Rings problem?

Question

let R be a ring with more than one element

If R has and identity element 1 for multiplication how can i show that 1 does not equal 0

Answer 1

The prior answer had the right idea, but was over-complicated. Here's what your prof is looking for:

If R has more than one element, pick an element A not equal to 0.

If 0 = 1, then 0 = 0*A = 1*A = A.

Contradiction. QED.

Answer 2

0 = 0
0 = 0+0 (additive identity)
0*a = (0+0)*a
0*a = 0*a + 0*a (distribuity)
a = a+a
0=a

Since this is true for all a in R, then R can only have one element and is the ring [{0},+,*}


So, 0=1 iff R has one element.