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I am working my way through Linear Algebra - Hoffman and Kunze. There is a very brief introduction to Modules in the chapter on Determinants. The authors state "..a module may be finitely generated without having a finite basis.". I am looking for an example for such a module.

I am not familiar with Group/Ring theory (although a know the basic definitions), so the examples else where are taking too long to understand.

For ex. A module without a basis forces me to go get an understanding of subgroupgs, cyclic groups, factor groups.

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It's funny because on page 165, they say "We repeat that a module may be ..." but they never actually said it before. – Auburn Jul 9 at 18:12
up vote 1 down vote accepted

Consider $\mathbb Z_n$ as a $\mathbb Z$ module.

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Ok. Annihilator of $\mathbb Z_n$ contains n. Thus it has no independent subset, let alone a basis. Thank you. – user869081 Jan 11 '13 at 16:09

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