 Solution

ID: past-exam-of-the-commutative-algebra-course-of-the-university-of-oxford/2023/2/d/solution

T = \prod_{n \geq 0} R,

with coordinatewise addition and multiplication and unit

(1, 1, \dots)

. Define the shift

g (a_{0}, a_{1}, a_{2}, \dots) = (a_{1}, a_{2}, a_{3}, \dots) .

Coordinatewise operations show that

g

preserves addition, multiplication and the unit, so it is a ring endomorphism. It is surjective because

(0, b_{0}, b_{1}, \dots)

maps to any prescribed

(b_{0}, b_{1}, \dots)

ker g = {(a, 0, 0, \dots) : a \in R} \neq = 0,

This ring fails the Noetherian hypothesis in c. Indeed, let

e_{n}

have a single

1

in coordinate

n

. The ideals

I_{n} = (e_{0}, \dots, e_{n})

form the strictly ascending chain

I_{0} ⊊ I_{1} ⊊ I_{2} ⊊ \dots,

since

e_{n + 1} \in / I_{n}

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