## Exhibit a nonzero torsion module over an integral domain whose annihilator is trivial

Exhibit an integral domain and a nonzero torsion module over such that . Prove that if is a finitely generated torsion -module ( a domain), then .

First consider the direct sum as a -module in the usual way. Every element of is annihilated by , where is the largest nonzero component of . However, for all , there exist elements not annihilated by . For example, if , then the element with in the component an 0 elsewhere is not annihilated by .

Suppose now that is a finitely generated torsion -module; say , with . For each , there exists a nonzero element such that , since is torsion. Let . Since is an integral domain, , and certainly . Thus .

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## Comments

Is there any difference between this exercise and 10.3.5 (https://crazyproject.wordpress.com/2011/04/15/every-finitely-generated-torsion-module-over-an-integral-domain-has-a-nonzero-annihilator/)?

Find a nonzero torsion module with trivial annihilator; prove that all finitely-generated torsion modules have non-trivial annihilator.

Kind of weird for D-F to include basically the same problem twice.

Yep, they’re the same. Actually there are a handful of other repeated exercises. When I recognize them I just link back- this one got by me.