Characterizing when $R[X]$ is integrally closed

Abstract:

Unlike the situation when dealing with integral domains, it is not always the case that the polynomial ring $R\left [ X \right ]$ is integrally closed when $R$ is an integrally closed commutative ring with nonzero zero divisors. In the main theorem it is shown that for an integrally closed reduced ring $R$, $R\left [ X \right ]$ is not integrally closed if and only if there exists a finitely generated dense ideal $J$ and an $R$-module homomorphism $s \in {\text {Ho}}{{\text {m}}_R}\left ( {J,R} \right )$ such that $s$ is integral over $R$ and $s$ is not defined by multiplication by a fixed element of $R$. As a corollary it is shown that $R\left [ X \right ]$ is integrally closed if and only if $R$ is integrally closed in $T\left ( {R\left [ X \right ]} \right )$, the total quotient ring of $R\left [ X \right ]$.