Sharp's Exercise 13.35

Exercise: Let $R$ be a subring of the commutative ring $S$ and suppose that S is integral over R.

i) Show that, if $r$ $\in$ $R$ is a unit in $S$ then $r$ is a unit in $R$.

ii) Show that, $Jac(R)=Jac(S)$ $\cap$ $R$

I need some hints for this two prove.

Firstly, i try to use integral definition for (i).

I know if we say that $s$ $\in$ $S$ integral over $R$ precisely when there exist $n$$\in$$N$ and $r_0,...,r_{n-1}$ $\in$ $R$ such that

$s^n+r_{n-1}s^{n-1}+\ldots+r_1s+r_0$ = 0

that is, if and only if $s$ is a root of a monic polynomial in $R[x]$.

But i dont know how can i show that (i) and (ii).

• general comment on formatting: if you wish to write subscripts, you better use {}: $r_n-1$ is without braces, $r_{n-1}$ is with braces around $'n-1'$. Jun 26, 2018 at 14:36

Let $s \in S$ be the inverse of $r$. $s$ is integral over $R$, hence there exists a relation with $a_0, \dots , a_{n-1} \in R$ $$s^n = -\sum_{i=0}^{n-1}a_i s^i$$ Multiplying everything by $r^{n-1}$ you get $$s = -\sum_{i=0}^{n-1}a_i r^{n-1-i} \in R$$ Proving (i).
For (ii) the inclusion $$Jac(R) \supseteq Jac(S) \cap R$$ is true for any extension of rings. The other inclusion should use (i), but I have to think about it.