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