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Let $k$ be a field and $k[a]$ an algebric extension. If $A$ is a reduced commutative algebra over $k[a]$ and $B$ is a subring which is an algebra over $k$, then is the following true: if there exist elements $x,y\in B$ such that $xa+y=0$ then $x=y=0$?

If it is not true in the general case, is it true in the case where $k[a]$ is an inseparable extension of degree $p$ over $k$, $A$ is finitely generated, and $B=A^{p}=\{{y^{p}\ |\ y\in A\}}$?

edited Jul 6 2011 at 18:52

GH
14.9k●1●31●53

asked Jul 6 2011 at 17:23

ventura
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What rôle does $B$ play in your first paragraph? – Mariano Suárez-Alvarez Jul 6 2011 at 17:36

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I might be missing something, but a counterexample to your first question is $A=B=k[a]$, $x=1$, $y=-a$. – GH Jul 6 2011 at 18:59

ventura · Answer 1 · 2011-07-06 19:09:08Z UTC

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im sorry i ment to write $x,y\in B$ i hope now its clear. $a\notin B$ thanks for the editing

edited Jul 6 2011 at 19:09

ventura

community wiki

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I fixed that, and corrected some other typos, grammar, and punctuation. Please give more care to your question next time if you want it to be taken seriously. – GH Jul 6 2011 at 18:54

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If you need $a\notin B$, a counterexample is $A=k(a)[T]$, $B=k[T]+Tk(a)[T]$, $x=T$, $y=-aT$. – Kevin Ventullo Jul 20 2011 at 22:15

sub ring of algebra over subfield

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