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Characteristics of rings

fredpeterson57

New member
Oct 15, 2019
1
Context: Let R be a unital ring. The characteristic of R is the smallest positive integer n such that $n\cdot 1=0$. If no such n exists, we say R has characteristic 0. We denote the characteristic of a ring by char(R).
I'm particularly lost as to how to prove the following propositions:
(a) Every unital ring of characteristic zero is infinite (I'm thinking of using a proof by contradiction for this, but I have no idea how)

(b) The characteristic of an integral domain is either 0 or prime (if I somehow manage to show that if the characteristic of an integral domain is composite or 1, then it is not an integral domain, then I think I will be able to prove this).
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,681
Context: Let R be a unital ring. The characteristic of R is the smallest positive integer n such that $n\cdot 1=0$. If no such n exists, we say R has characteristic 0. We denote the characteristic of a ring by char(R).
I'm particularly lost as to how to prove the following propositions:
(a) Every unital ring of characteristic zero is infinite (I'm thinking of using a proof by contradiction for this, but I have no idea how)

(b) The characteristic of an integral domain is either 0 or prime (if I somehow manage to show that if the characteristic of an integral domain is composite or 1, then it is not an integral domain, then I think I will be able to prove this).
(a) Show that if the ring has characteristic zero then the elements $n\cdot 1\ (n\in\Bbb{N})$ are all different.

(b) If the characteristic $n$ of the ring is a composite number, say $n = pq$, then $0 = n\cdot1 = pq\cdot1 = (p\cdot1)(q\cdot1)$. Now use the fact that an integral domain does not have zero-divisors to show that either $p\cdot1=0$ or $q\cdot1=0$ (contradicting the fact that $n$ is the smallest number with that property).