Does anyone have a proof for The Proposition about a field having characteristic p ( a prime) or 0?

Answer 1

-If R is a ring such that (R\{0},*) is a group, then R is called a division ring.

- If R is a division ring and is commutative, then R is called a field

-If R is a division ring, then R has no divisors of zero.
Let a,b be elements of R with a =/= 0 and suppose ab=0. Now a=/=0 => x in R such that ax=1. Now
0 = x0 = x(ab) = (xa)b = 1b = b and, in a similar fashion, ab=0 with b =/= 0 => a=0 so R has no divisors of 0.

-Let R be a ring. If there is a least positive integer n such that nr=0 for all r in R, then n is called the characteristic of R.

-Let R be a ring with more than one element and no divisors of 0. The characteristic of R is then either prime or 0.
Suppose that char(R)=n=/=0 and n=mk where 1 m(kr)=0. In either case, there is a contradiction since both m and k are less than n. Thus n must be prime.

Hope that helps

Doug