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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-2/1e/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 1E / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

The zero element is torsion. If

m, n \in T

, choose nonzero

r, s \in R

with

r m = 0

and

s n = 0

. Since

R

is an integral domain,

rs \neq = 0

, and

rs (m + n) = s (r m) + r (s n) = 0.

Thus $m + n \in T$ . If $a \in R$ , then $r (am) = a (r m) = 0$ , so $am \in T$ . Hence $T$ is closed under addition and scalar multiplication and is an submodule of $M$ .

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