 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2005/algebra/3/ii/solution

Let

J = ⋃_{n} I_{n}

. It contains zero. If

a, b \in J

, both belong to

I_{N}

for some common

N

, so

a - b \in I_{N} \subseteq J

. If

r \in R

, then

r a \in I_{N} \subseteq J

. These are the ideal axioms, proving

J

is an ideal.

J = (g)

. Since

g \in J

, there is

n_{0}

with

g \in I_{n_{0}}

. For

n \geq n_{0}

(g) \subseteq I_{n_{0}} \subseteq I_{n} \subseteq J = (g) .

All inclusions are equalities. Hence the chain stabilizes at $I_{n_{0}}$ . This includes

J = (0)

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