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\begin{document}

\title{Math 172 (Su):  HW number 11}
\maketitle

\vfill
{\bf 1.} Let $I=\left< x^2 y^2, x^3 y \right>$ be an ideal in $K[x,y]$.

(a) In the $(m,n)$-plane, plot the set of exponent vectors $(m,n)$ of
monomials $x^m y^n$ that appear as elements of $I$.

(b) If we apply the division algorithm to an element $f \in K[x,y]$
using the generators of $I$ as divisors, what terms can appear in the
remainder?


\vfill


{\bf 2.}
Complete the last part of the proof of Dickson's lemma:

Let $I = \left< x^\alpha : \alpha \in A \right>$ be a monomial ideal,
and suppose there is a finite basis for $I$ such that 
$$I = \left< x^{\beta(1)}, x^{\beta(2)}, ..., x^{\beta(s)} \right>.$$

Prove that there is a finite basis for $I$ from among the original
generating set:
$$I = \left< x^{\alpha(1)}, x^{\alpha(2)}, ..., x^{\alpha(s)} \right>$$
where each $\alpha(i) \in A$.


\vfill

{\bf 3.} Show that the ring $K[x_1,...,x_n]$ is Noetherian.

(Recall that at the beginning of this class we defined 
a {\em Noetherian} ring to be a ring in which there is no
infinite ascending chain of ideals: i.e., in any chain of ideals 
$$
I_1 \subseteq
I_2 \subseteq
I_3 \subseteq
\cdots
$$
there is an $n$ beyond which $I_k=I_n$ for all $k\geq n$.)

\bigskip
Hint: use the Hilbert Basis Theorem that says that 
every ideal in this ring is finitely generated.

\bigskip 
\bigskip 
(Note: the book defines Noetherian differently:
as a ring in which every ideal is finitely generated.  
As it turns out, the two definitions are equivalent, 
and in this problem you are showing one half of this equivalence: 
that the book's definition implies our definition of Noetherian.)

\end{document}