Answer to Question #160678 in Abstract Algebra for DHRUV PANDIL

Question #160678

Let f is a field. Then prove that (0) is a prime ideal in fm


1
Expert's answer
2021-02-04T07:34:59-0500

We will prove that (0)(0) is the maximal proper ideal and this will imply that it is prime. Suppose that (0)(0) is not maximal, then m,(0)m\exists \mathfrak{m}, (0)\subsetneq \mathfrak{m} a maximal ideal. Then, am,a0\exists a\in \mathfrak{m}, a\neq 0, but this implies that bF,bm,\forall b \in F, b\in \mathfrak{m}, as (ba1)am(b\cdot a^{-1})\cdot a \in \mathfrak{m} and thus bmb\in \mathfrak{m} (FF is a field so for any non-zero element there is an inverse element). Therefore m=F\mathfrak{m} = F and so it is not proper. So we conclude that (0)(0) is a maximal proper ideal and thus is prime.


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