\magnification=1200

\input AMSSYM.DEF
\def\Q{\mathord{\Bbb Q}}
\def\R{\mathord{\Bbb R}}
\def\N{\mathord{{\Bbb N}}}
\def\Z{\mathord{{\Bbb Z}}}

\def\monoid{*monoid}
\def\supp{\mathop{{\rm supp}}}
\def\tmo{\mathrel{\preceq}}
\def\stmo{\mathrel{\prec}}

\def\graver{\mathop{\rm Gr}}
\def\mb{\mathop{\rm MinBal}}


\noindent {\bf Lecture 4. }

Recall $I\subseteq M$ is an ideal if $I+M\subseteq I$.  $I$ is said to be {\it prime\/} if its complement is closed under $+$.  

\bigskip
\noindent {\bf Fact.}  Ideal $I\subseteq \N^n$ is prime if and only if its minimal elements are among the generators of $\N^n$.  {\it Proof\/}.  If a minimal element of $I$ is not a generator of $\N^n$, it can be expressed as a sum of two elements not in $I$.

\bigskip
\noindent It follows that there is a bijection:
$$\hbox{subsets of $\{\,1,2,\ldots,n\,\}$}\;\longleftrightarrow \hbox{prime ideals of $\N^n$}$$

\end