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The characteristic of an integral domain is prime or zero

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.28

Solution: Suppose the characteristic $n$ of $R$ is composite, and that $n = ab$ where $a$ and $b$ are both less than $n$. Letting $$\varphi : \mathbb{Z} \rightarrow R$$ be the ring homomorphism that takes $k \in \mathbb{Z}$ to the $k$-fold sum of $1$ or $-1$, we have $\varphi(a)$ and $\varphi(b)$ nonzero. However, $$\varphi(a)\varphi(b) = \varphi(ab) = \varphi(n) = 0,$$ so that $\varphi(a)$ and $\varphi(b)$ are zero divisors. Thus we have a contradiction.

Hence, the characteristic of $R$ is not composite, and thus must be a prime or zero.


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