Intereting Posts

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It’s well known that if $K$ is a finitely generated extension of some field $E$, then any intermediate field $F$, $E\subseteq F\subseteq K$, is also finitely generated over $E$.

I’m curious, does the same hold for rings?

Say $S$ is a ring, and $R\supset S$ is a finitely generated extension of $S$. If $T$ is any intermediate ring, is it necessarily true that $T$ is finitely generated over $S$ as a ring?

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- Let G be a finite group with more than one element. Show that G has an element of prime order

Is it as simple as saying that for any $t\in T$, $T$ can be generated by the generators of $R$ over $S$? I feel unsure about this statement, since it’s not clear to me that the generators of $R$ over $S$ need be in $T$.

If not, what is an example what shows otherwise? Thanks.

- Endomorphism Rings of finite length Modules are semiprimary
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- the the polynomial $K$-algebra $K$ is not local

Let $k$ be a field. Then $k[x, y]$ is a finitely generated extension of $k$, yet the subring $R\subseteq k[x,y]$ generated by $\{xy^i:i\geq0\}$ is not finitely generated.

Life would be very much simpler in some respects if the answer were yes ðŸ™‚

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