Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $M$ be a module, finitely generated,over the principal ideal domain $D$. If $D$ is a field, then $M$ has a basis. But if $D$ is not a field theis is not truth.

Can you give me an example of a module over the integers $\mathbb Z$ finitely generated without a base?

share|improve this question

1 Answer 1

up vote 5 down vote accepted

Any finitely-generated torsion abelian group will do (and all such groups are actually finite). All elements are annihilated by some nonzero integer, and hence you have no linearly independent sets.

share|improve this answer
1  
Notice that torsion is a requirement here. Because finitely generated torsion-free modules over PIDs are free. –  JSchlather Feb 23 '13 at 16:03

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.