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**JJMC89** (a) Suppose that $\displaystyle A$ is an integral domain of positive characteristic. Suppose that $\displaystyle I \subsetneq A$ is an ideal. Prove that $\displaystyle A/I$ has the same characteristic as $\displaystyle A$.

Hints - Try to show the following:

1) The characteristic of any integral domain is either zero or a prime

2) If for an abelian group G we define $\displaystyle exponent(G):=Exp(G):=\max\{n\in\mathbb{N} | n=ord(x)\mbox{ for some }x\in G\}$ , then

the exponent of any subgroup of G is a divisor of the group's exponent or zero if the

latter is zero.

(b) Give an example which demonstrates that $\displaystyle A/I$ need not be an integral domain.