If $S$ is a free $R$-algebra, then no element of $S$ is integral over $R$? And other questions.

Question

Suppose $R$ and $S$ are commutative rings.

1. If $S$ is any $R$-algebra (not necessarily finitely generated), then every element of $S$ is written as a polynomial with coefficients in $R$ ?

2. If $s \in S$ is integral over $R$, then there exists a monic polynomial over $R$, say $p(x) \in R[x]$, such that $p(s)=0$. So, if $S$ is a free $R$-algebra, then no element of $S$ is integral over $R$?