Let f is a field. Then prove that (0) is a prime ideal in fm
We will prove that is the maximal proper ideal and this will imply that it is prime. Suppose that is not maximal, then a maximal ideal. Then, , but this implies that as and thus ( is a field so for any non-zero element there is an inverse element). Therefore and so it is not proper. So we conclude that is a maximal proper ideal and thus is prime.
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