I'm trying to motivate the notion of integrality in a ring extension. It seems that the following would be a good motivation, because it would show that the notion of algebraic elements over a ring is not useful.
Here's the thing I believe is true: Let $R\subseteq S$ be a ring extension. The set of elements of $S$ that are algebraic over $R$ (i.e. satisfy a polynomial equation with coefficients in $R$) is not necessarily a ring.
But I can find no discussion of this point on the internet. Can someone provide a concrete example? Thanks.