 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ib/paper-3/10e/a/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Suppose

I

is prime. If

(a + I) (b + I) = 0 + I,

then

ab \in I

, so

a \in I

b \in I

. Thus one factor is zero and

R / I

is an integral domain.

Conversely, if

R / I

is an integral domain and

ab \in I

, then

(a + I) (b + I) = 0 + I,

a + I = 0 + I

b + I = 0 + I

. Therefore

a \in I

b \in I

, and

I

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