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Let $R$ be commutative ring with identity that contains a field $K$ as a subring. If$ $R is a finite dimensional vector space over the field $K$, prove that every prime ideal in $R$ will be maximal.

My idea was to prove the integral domain $R/p$ (if $p$ is a prime ideal) was a field as for any ideal $P$, $R/P$ will be a field iff $P$ is maximal in $R$. But how can I use the fact that $R$ is finite dimensional over $K$? I don't understand.

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  • $\begingroup$ An integral domain that is also a finite dimensional vector space over a field is always a field. Searching...1 $\endgroup$ Jun 30, 2018 at 16:00
  • $\begingroup$ I suspect older (and possibly more canonical) dupe targets exist. Still searching... $\endgroup$ Jun 30, 2018 at 16:03
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    $\begingroup$ This is IMO a better dupe target. Fixing... $\endgroup$ Jun 30, 2018 at 16:05
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    $\begingroup$ Oops. I forgot that I can no longer change dupe targets on the fly :-). Anyway, this was my first find. Also covers this, but I tend to prefer the older version. $\endgroup$ Jun 30, 2018 at 16:08

2 Answers 2

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$A:=R/p$ is also finite dimensional and has no zero divisors.
Take any nonzero $a\in A$ and consider its powers $1,a, a^2,\dots$, these are linearly dependent. Take the least $n\in\Bbb N$ giving $\lambda_na^n+\dots+\lambda_0=0$.

Since we can cancel out $a$, the constant term $\lambda_0$ is nonzero.
Then dividing by $-\lambda_0$ and pulling out $a$, we arrive to $a\cdot f(a)=1$ for a specific polynomial $f$, yielding an inverse for $a$.

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Hint:

Note that $R/\mathfrak p$ has finite dimension over $K$. You have to prove that any non-zero element $x$ in $R/\mathfrak p$ is invertible. Consider the map $$R/\mathfrak p\xrightarrow {\enspace{}\times x\enspace\:}R/\mathfrak p.$$ This is an endomorphism of the $K$-vector space $R/\mathfrak p$, and it is injective since $R/\mathfrak p$ is an integral domain.

Note : The more general following result is often used:

Let $A\subset B$ two integral domains, such that $B$ is a finitely generated $A$-module. Then $A$ is a field if and only if $B$ is.

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