Integral elements subalgebra of an arbitrary ring map

Let $R\to S$ be a unital ring map of commutative rings, with $\phi$ its underlying map, and let $\phi (R)\subseteq S$ be its image. An element $s\in S$ is integral over the ring map $R\to S$ when $s$ is a root of a monic polynomial with coefficients in the image of $R$ (Integral elements over a commutative ring and algebraic integers): that is, when

$s^d+\phi (a_{d-1})s^{d-1}+\cdots +\phi (a_1)s+\phi (a_0)=0$

for some $d\ge 1$ and some $a_0,\dots ,a_{d-1}\in R$. This is integrality of the element $s$ over $\phi (R)$. The ring map $R\to S$ is integral in the sense of Integral ring maps and integral extensions precisely when every element $s\in S$ satisfies such an equation.

Write

${\mathrm{Int}}_R(S)=\{s\in S:s\text{ is integral over }R\to S\}$

for this set. Then ${\mathrm{Int}}_R(S)$ is a subring of $S$ (Subring: a subset containing $1_R$ and closed under addition, additive inverses and multiplication) containing $\phi (R)$, hence an $R$-subalgebra of $S$ for the restricted structure map (Algebras over a commutative ring, central structure maps, and algebra homomorphisms); it is called the integral closure of the image of $R$ in $S$, and in this page the notation $S^{\mathrm{\prime }}\subseteq S$ always denotes this relative integral closure. The reason is Integral elements over a nonzero base ring form a subring: if $\phi (R)\mathrm{\ne }0$ the published statement applies to the inclusion $\phi (R)\subseteq S$, whose integral elements over $\phi (R)$ are exactly the elements listed above; if $\phi (R)=0$ then $1_S=\phi (1_R)=0$, so $S=0$ and ${\mathrm{Int}}_R(S)=\{0\}=S$ is again a subring.

Conventions kept here. (i) No hypothesis of injectivity is imposed: the map $R\to S$ may have a kernel, and ${\mathrm{Int}}_R(S)$ is a subring of $S$ containing the image, not of $R$. (ii) No hypothesis excluding zero divisors or nilpotents is imposed, and all arguments below proceed in $S$ itself; in particular Integrality and integral closure commute with localisation may be applied through the structure map without assuming that $R\to S$ is injective. (iii) When $R\subseteq S$ are domains in the sense of Integral closure in an extension ring and integrally closed domains, this set is the integral closure of $R$ in $S$, so the present definition specialises to the published one, which is not assumed here.