Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 9E / b

Solution

The coefficient condition says precisely that every nonconstant monomial has positive powers of both variables. Thus

R = K + X Y K [X, Y] .

c + X Y f

and

d + X Y g

lie in

R

, so do their sum and their product

c d + X Y (c g + df + X Y f g),

R

is a subring of

K [X, Y]

For

n \geq 1

, let

I_{n} = (X Y, X Y^{2}, \dots, X Y^{n})_{R} .

Then

I_{n} \subseteq I_{n + 1}

. The containment is strict because

X Y^{n + 1} \in / I_{n}

: multiplying a generator

X Y^{j}

by an element of

R

produces either a scalar multiple of

X Y^{j}

, or terms divisible by

X^{2}

. It cannot produce the monomial

X Y^{n + 1}

when

j \leq n

. Hence

I_{1} ⊊ I_{2} ⊊ I_{3} ⊊ \dots

is a strictly increasing ideal chain. The constant-plus-ideal non-Noetherian subring therefore satisfies

R is not Noetherian .

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