Question

ote. lhere Is 1o at the end.) For a vector space over a field, we know that any linearly independent set can be extended to a basis, and that any spanning set can be reduced to a basis. Let M be a free module of finite rank over a PID R. Are the corresponding statements for M valid? Give proofs or counterexamples.

Answer 1

Consider the module $\Bbb Z$ over $\Bbb Z$.

Note that $\Bbb Z$ is a PID as it is generated by $[1]$.

Also rank of $\Bbb Z$ over $\Bbb Z$ is $1$ as $\{1\}$ is a basis of $\Bbb Z$ over $\Bbb Z$.

Consider the set $\{2,3\}$.

Since $\gcd(2,3)=1$ so $\{2,3\}$ is a generating set of $\Bbb Z$.

But $\{2,3\}$ cant be reduced to a basis.--------------(1)

Again the set $\{2\}$ is linearly independent but it cant be extended to a basis because we cant write every element of $\Bbb Z$ as a linearly combination of $2$.---------------------------(2)

Thus (1) and (2) provide examples where the above statements are not true.