We have a ring $R$, commutative, and $f_1,\dots,f_n$ polynomials in $R[x]$ monic, with $\deg f_i\ge 1$. It is straightforward to show that there is a ring extension $R\subset S$ such that $S$ contains all of the roots of the set of polynomials (the polynomials split in $S$). Basically one uses the result that if $x$ is integral over $R$, then $R[x]$ is a finitely generated as an $R$-module, and then induct.
I would like to
a) Show that there is an integral ring extension $T$ over $R$ such that every monic polynomial (non-constant) splits in $R$.
and use a) to show
b) That if we have $R\subset T\subset S$, with $T$ the integral closure of $R$ in $S$, then for any $f,g\in S[x]$ monic, with $fg\in T[x]$, then $f$ and $g$ are each in $T[x]$. [No assumptions about the integrality of the ring $R$.]
If we could assume that $R$ were integral it would be easier, since we could think of things in the field of fractions. Any help for how to proceed would be much appreciated.