Question

Abstract algebra
given finite ring R with unity 1 not equal to 0, and R has no zero divisors.
prove R is a division ring.

Answer 1

so if R is finite, and has n elements, then I can take some element \(a\ne0\) in ring and take the left product with and each of the n-1 non zero element, and get back n-1 elements(each will be different because we have cancellation). So since we get back n-1 elements one of them must be 1
So I have shown there exists \(a_1\) st \(a*a_1=1\)

now we can do the same thing again to show there is some \(a_2 \ s.t. \ a_2*a=1\)


how do i show that \(a_1=a_2\)?

Answer 2

@oldrin.bataku when you get a sec, could you look at this. please sir

Answer 3

@ChristopherToni that goes for you as well:)

Answer 4

let me see

Answer 5

every place I see this proof, they stop here. no one ever shows that the left and right elemets are the same...

Answer 6

>we have cancellation

what do you mean here?

Answer 7

ab=cb imples a=c

Answer 8

we get that for free because there are no zero divisors

Answer 9

sorry i forgot to add that to the question.

Answer 10

I edited it.

Answer 11

ah there are no zero-divisors... good good

Answer 12

sorry about that

Answer 13

ok got it

Answer 14

\(a_1=1\cdot a_1=(a_2a)a_1=a_2(aa_1)=a_2\cdot1=a_2\)

Answer 15

ahh lol, wow. I thought I needed to some up with a bijection from R to R and do some ugly stuff. ty again:)

Answer 16

some = come

Answer 17

NP :-p glad to help

Answer 18

impressive as always

Answer 19

do you know how long it takes to reach \(90\) from \(89\) SmartScore-wise? it is automatic, yeah?