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\LARGE
\centerline{\bf Monomial Subrings and Systems of Subintegrality}
\Large

\medskip
\centerline{Leslie G. Roberts}

\bigskip
\noindent{\bf Definition 1.}
Let $A\subseteq B$ be commutative rings. Then $A\subseteq B$ is subintegral
$\Leftrightarrow$ 
\begin{enumerate}
\item[(i)] $B$ is integral over $A$.
\item[(ii)] The induced map ${\mbox {\rm Spec}}B\rightarrow {\mbox {\rm Spec} A}$ 
is a bijection and the induced homomorphism 
\[ \kappa(\frac{A}{Q\cap A})\rightarrow \kappa(\frac{B}{Q})\]
is an isomorphism for all $Q\in {\rm Spec}B$.
\end{enumerate}

\noindent {\bf Definition 2.}
Let $A\subseteq B$ be commutative rings. Then $A\subseteq B$ is weakly 
subintegral
$\Leftrightarrow$ 

\begin{enumerate}
\item[(i)] $B$ is integral over $A$.
\item[(ii)] The induced map ${\mbox {\rm Spec}}B\rightarrow {\mbox {\rm Spec} A}$
is a bijection and the induced homomorphism 
\[ \kappa(\frac{A}{Q\cap A})\rightarrow \kappa(\frac{B}{Q})\]
is a purely inseparable field extension for all $Q\in {\rm Spec}B$.
\end{enumerate}

\pagebreak
Obviously subintegral $\Rightarrow$ weakly subintegral, and if $\mathbb{Q}\subset A$ then subintegral $\Leftrightarrow$ weakly subintegral.

\medskip
\noindent{\bf Examples 3.}
\begin{enumerate}

\item  $k[t^2,t^3]\subseteq k[t]$ is subintegral. ($k$ any commutative ring.)

\vskip 15pt
\centerline{\includegraphics[bb=40 134 200 254, scale=.3]{cusp.ps}
$\leftarrow$ \includegraphics[bb=70 134 100 256, scale=.32]{line.ps}
            }

\vskip 35pt
\item  $\mathbb{F}_2[t^2] \subset \mathbb{F}_2[t]$ is weakly subintegral
but not subintegral.

\end{enumerate}

\noindent{\bf Facts 4.} Given an arbitrary extension $A\subset B$ of commutative rings
\begin{enumerate}
\item There is a largest subextension ${}_B^+A$, $A\subseteq {}^+_BA \subseteq B$ such that $A\subseteq {}^+_BA$ is subintegral. ${}^+_BA$ is called the seminormalization of $A$ in $B$, and $A$ is seminormal in $B$ if
 ${}^+_BA = A$.
 \item There is a largest subextension ${}^*_BA$, $A\subseteq {}^*_BA \subseteq B$ such that $A\subseteq {}^*_BA$ is weakly subintegral. ${}^*_BA$ is called the weak normalization of $A$ in $B$, and $A$ is weakly normal in $B$ if
 ${}^*_BA = A$.
\pagebreak
\item An elementary subintegral extension of commutative rings is an
extension of the form 
$R\subseteq R[x]$ where $x^2, x^3\in R$. Then we have
that $x\in {}_B^+A \Leftrightarrow $ there is a sequence of extensions
$A=A_0\subset A_1\subset \cdots \subset A_d$ with $x\in A_d$ and
$A_{i-1}\subset A_i$ elementary subintegral ($1\leq i\leq d$).
\item An elementary weakly subintegral extension is an extension of the form
$R\subseteq R[x]$ where $x^p, px\in R$ for some prime $p$. Then we have
that $x\in {}_B^*A \Leftrightarrow $ there is a sequence of extensions
$A=A_0\subset A_1\subset \cdots \subset A_d$ with $x\in A_d$ and
$A_{i-1}\subset A_i$ either elementary subintegral or elementary
weakly subintegral ($1\leq i\leq d$).
\end{enumerate}
\medskip
The extension $\mathbb{Z}[t^2,2t] \subset \mathbb{Z}[t]$  is elementary weakly subintegral. Motivated by such examples we introduce the following definition:

\medskip
\flushleft{\bf Definition 5.} A monomial subring $A$ of $B=\mathbb{Z}[t]$ is a
subring generated as an abelian group by ``monomials'' $at^i$ ($a\in \mathbb{Z},\ i\geq 1$).

\medskip
As an abelian group $A=\mathbb{Z}\oplus(\oplus_{i\geq 1}\mathbb{Z}b_it^i)$ 
for integers $b_i\geq 0$. For simplicity assume that $t^d\in A$ for some 
$d\geq 1$. Without loss of generality we may assume that $b_i\neq 0$ for $i$ sufficiently large. It is convenient to describe $A$ by giving its {\em coefficient
sequence} $1,b_1,b_2,\ldots$. 

\pagebreak

{\bf Easy Observations 6.}
\begin{enumerate}
\item If $c_i$ is an arbitrary sequence of non-negative integers (with $c_0=1)$ then $\mathbb{Z}\oplus(\oplus_{i\geq 1}\mathbb{Z}c_it^i)$ is a ring if
and only if $c_{i+j}\mid c_i c_j$. 

\item The above condition can be checked at each prime, so that if $q$ is a 
prime and $c^{[q]}$ is the $q$-primary part of an integer $c$ then 
$C=\mathbb{Z}\oplus(\oplus_{i\geq 1}\mathbb{Z}c_it^i)$ is a ring if and
only if $C^{[q]} = \mathbb{Z}\oplus(\oplus_{i\geq 1}\mathbb{Z}c_i^{[q]}t^i)$
is a ring for all primes $q$. 

\item The sequence $\{ b_i\}$ becomes periodic if $i$ is sufficiently large.
Explicitly there are integers $N$ and $\delta$ such that if $i\geq N$
then $b_{i+\delta}=b_i$. (Take $\delta$ to be the smallest such period.)
\item  If we obtain the sequence $a_i$ by extending $b_i$ back to $i=0$
periodically (i.e. $a_i = b_{i+n \delta}$ for $n>>0$) then
$\mathbb{Z}\oplus(\oplus_{i\geq 1}\mathbb{Z}a_it^i)$ is a ring
(which we will call the {\em stabilization} of $A$). The stabilization of
$A$ is a monomial subring of $\mathbb{Z}[t]$ which is larger that $A$.

\item If we let $\overline a$ denote the ``squarefree'' part of an integer
$A$ then $\overline A = \mathbb{Z}\oplus(\oplus_{i\geq 1}\mathbb{Z}\overline a_it^i)$ is a ring.
\end{enumerate}
\pagebreak
{\bf Theorem 7.} If $A$ is a monomial subring of $B=\mathbb{Z}[t]$ and 
$M=at^i$ then

{\rm (a)} $A[M]$ is subintegral over $A$ if and only if $M^i\in A$ for 
$i>>0$.

{\rm (b)} $A[M]$ is weakly subintegral over $A$ if and only if there is an integer valued polynomial $f\in \mathbb{Q}$ such that $f(0)=1$ and $f(i)M^i\in A$ for $i>>0$. 

\medskip
Our goal is to determine the seminormal and weakly normal monomial subrings 
of $\mathbb{Z}[t]$ (which contain $t^\delta$ for some integer $\delta\geq 1$). From the above observations it follows that $A$ is 
seminormal (respectively weakly normal) in $\mathbb{Z}[t]$ if and only
if $A^{[q]}$ is seminormal (respectively weakly normal) in $\mathbb{Z}[t]$. 
Thus it suffices to consider the case where all the coefficients are
powers of one prime $q$. Then we have

\medskip
{\bf Theorem 8.}  Let $A$ be a monomial subring of $\mathbb{Z}[t]$
and suppose that all the coefficients of $A$ are powers of one prime $q$.
Then 

{\rm (a)} $A$ is seminormal in $\mathbb{Z}[t]$ if and only if the coefficient
sequence of $A$ is of the form $1, q, q, \ldots,\ q, 1, \ldots$
with some period $\delta\geq 1$. 

{\rm (b)} $A$ is weakly normal in $\mathbb{Z}[t]$ if and only if the 
coefficient sequence of $A$ is  
$1, q, q, \ldots, q, 1, \ldots$ with some period $\delta\geq 1$, GCD$(\delta,q)=1$.

\pagebreak
{\bf Theorem 9.} 
 Let  $A=\mathbb{Z}\oplus(\oplus_{i\geq 1}\mathbb{Z}b_it^i)$ be an arbitrary monomial subring of $B=\mathbb{Z}[t]$. Then ${}^+_BA = \overline C$, where $C$ is the stabilization of $A$, and  ${}^*_BA$ is obtained by 
first forming $D={}^+_BA$ and then for each prime $q$ replacing $D^{[q]}$
 (with coefficient sequence $1, q, \ldots , q, 1, \ldots$ and
period $\delta_q$) by the corresponding ring with period $\delta_q/\delta_q^{[q]}$.

\medskip
{\bf Examples 10.} {\rm (a)} Let $A=\mathbb{Z}[6t,3t^2,2t^3,3t^4,t^6]$ with coefficient sequence $1, 6, 3, 2, 3, 6, 1 \ldots$ of period 6.
Then $A^{[2]}$ has coefficient sequence $1, 2, 1 \ldots$ repeating with period 2 and $A^{[3]}$ has coefficient sequence $1, 3, 3, 1 \ldots$ of period 3
(i.e. $A^{[2]}=\mathbb{Z}[2t,t^2]$ and $A^{[3]} = \mathbb{Z}[3t,3t^2,t^3]$). 
Both $A^{[2]}$ and $A^{[3]}$ are seminormal so $A$ is seminormal in $\mathbb{Z}[t]$. However ${}^*_BA=\mathbb{Z}[t]$.
  
{\rm (b)} Let $A=\mathbb{Z}[12t,6t^2,t^3]$. Then $A$ is already stable and
$D={}^+_BA = \mathbb{Z}[6t,6t^2,t^3]$. Here $D^{[2]}$ has coefficient
sequence $1\ 2\ 2\ 1\ldots $ (with period 3). $D^{[2]}$
is already weakly normal in $\mathbb{Z}[t]$. However $D^{[3]}$ has
coefficient sequence $1\ 3\ 3\ 1\ldots $ (with period 3). 
This is weakly subintegral in $\mathbb{Z}[t]$ so
${}^*_BA=\mathbb{Z}[2t,2t^2,t^3]$.

{\rm (c)} $A = \mathbb{Z}[6t,2t^2,3t^3,2t^4,t^6]$ (with coefficient
sequence $1, 6, 2, 3, 2, 6, 1 \ldots$) is weakly normal in $\mathbb{Z}[t]$.

{\rm (d)} $A=\mathbb{Z}[t^3,3t^4, 6t^5]$ has coefficient sequence
$1, 0, 0, 1, 3, 6, 1, 3, 3, 1, \ldots$ (now periodic of period 3). This 
stabilizes to $1\ 3\ 3\ 1$  repeating with period 3, so
${}^+_BA=\mathbb{Z}[3t,3t^2,t^3]$ and ${}^*_BA=\mathbb{Z}[t]$.

\bigskip
{\bf Characterization of weak subintegrality.}

\medskip
For any commutative ring $R$ define a multiplication of $R[[T]]$ by
$T^i*T^j = \binom{i+j}{i}T^{i+j}$. It $r\in R$ define Exp$(r)$ by
Exp$(r) = 1+\sum_{i=1}^\infty r^i T^i$. Then $\hbox{ Exp}(r+s) = \hbox{Exp}(r)*\hbox{Exp}(s)$. 

{\bf Theorem 11.} Let $A\subseteq B$ be an arbitrary extension of commutative
rings and let $b\in B$. Then $b\in {}^*_BA$ if and only if there is a 
polynomial $f\in B[T]$ with $f(0)=1$ such that $f(T)*\hbox{Exp}(r)\in A[[T]]$. 

\medskip
{\bf Problem 12.} Is there a similar characterization of subintegrality,
in the case when $A$ does not contain $\mathbb{Q}$?

\medskip
{\bf Example 13.} Suppose that $b^i\in A$ for $i\geq n$. Let $f(T)=
1+\sum_{i=1}^{n-1} (-1)^i b^i T^i$. Then $f(T)*\hbox{Exp}(b) = 1+$ terms in
$b^i, i\geq n$ so $f(T)*\hbox{Exp}(b)\in A[[T]]$.  

Explicitly $f(T)*\hbox{Exp}(b) = 1+\sum_{i=n}^\infty(-1)^{n-1}\binom{i-1}{n-1}b^iT^i$. Thus in order to have $f(T)*\hbox{Exp}(b)\in A[[T]]$ 
(and hence that $b\in {}^*_AB$) we
require only that $\binom{i-1}{n-1}b^i\in A$. For example take  
$A=\mathbb{Z}[\{{\binom{i-1}{n-1}t^i\}}_{i\geq n}]$ and $B=\mathbb{Z}[t]$. 
In this case  $A \subseteq B$ is weakly subintegral. 

\medskip
{\bf Example 14.} Let $n=3$. Then
$A=$ $\mathbb{Z}[t^3, 3t^4, 6t^5, 10t^6 \ldots] = \mathbb{Z}[t^3, 3t^4, 6t^5]$
which has coefficient sequence $1,0,0,1,3,6,1,3,3,1,\ldots$ and
stabilization sequence $1,3,3,1,\ldots$ , corresponding to
$\mathbb{Z}[3t,3t^2,t^3]$ (as in Example 10d).

\medskip
We wish to determine the stabilization sequence of $A=\mathbb{Z}[\{{\binom{i-1}{n-1}t^i\}}_{i\geq n}]$ for all $n$. First a preliminary theorem:

\medskip
{\bf Theorem 15.} Let $n$ and $i$ be a fixed integers, $n\geq 1$ and $1\leq i\leq n$. Then \{GCD$\{ \binom{d-1}{n-1} \}\mid {d\equiv i\ {\rm MOD}\ n}\} = n/\hbox{GCD}(n,i)$.  

\medskip
{\bf Examples 16.} If $n=3$ then the sequence $\{n/\hbox{GCD}(n,i)\}$ (as $i$
varies) is the sequence $1,3,3,1,3,3,1,\ldots$ . If $n=6$ then
$\{n/\hbox{GCD}(n,i)\}$ is the sequence $1,6,3,2,3,6,1,\ldots$ . 

\medskip
It follows from the Theorem and a not too difficult calculation that
the stabilization of $A=\mathbb{Z}[\{\binom{i-1}{n-1}t^i\}_{i\geq n}]$ is
$\mathbb{Z}[\{(n/\hbox{GCD}(i,n))t^i\}_{1\leq i\leq n}]$ which has
stabilization sequence $\{n/\hbox{GCD}(i,n)\}_{i\geq 0}$.   
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\bigskip
\noindent {\bf Exercise 17.} Give an example of an extension $A\subseteq B$
that satisfies 1(ii) but not 1(i).

\pagebreak

\bigskip
Most of the above material is found in \cite{rr99} (available
on my web page). Basic references
are \cite{swan} for subintegrality and \cite{yanag85} for weak 
subintegrality. Theorem 11   is found in \cite{rrs96}.

%bibliography generated by bibtex
\begin{thebibliography}{1}

\bibitem{rr99}
Les Reid and Leslie~G. Roberts.
\newblock Monomial subrings and systems of subintegrality.
\newblock To appear in {\it JPAA}.

\bibitem{rrs96}
Les Reid, Leslie~G. Roberts, and Balwant Singh.
\newblock On weak subintegrality.
\newblock {\em JPAA}, 114:93--109, 1996.

\bibitem{swan}
R.~G. Swan.
\newblock On seminormality.
\newblock {\em J.Algebra}, 67:210--229, 1980.

\bibitem{yanag85}
H.~Yanagihara.
\newblock On an intrinsic definition of weakly normal rings.
\newblock {\em Kobe J. Math.}, 2:89--98, 1985.

\end{thebibliography}
 
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