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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 $B=A^{p}=\{{y^{p}\ |\ y\in A\}}$?

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What rôle does $B$ play in your first paragraph? –  Mariano Suárez-Alvarez Jul 6 '11 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 from MO Jul 6 '11 at 18:59
    
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 '11 at 22:15 –  Todd Trimble Oct 19 '13 at 23:35

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