Originally Posted by

**Simply82358** Let K be a field, and (x) -> K[x] be an injective map where (x) is the principal ideal generated by x.

(i) Prove that the quotient K[x]-module, K[x]/(x), is isomorphic to K as an abelian group and also as a K-vector space.

Hence there is an exact sequence of K[x]-modules :

0 -> (x) -> K[x] -> K -> 0

(ii) Is this sequence split as a sequence of K[x]-modules ?

(iii) Is this sequence split as a sequence of K-modules ?

(i) is clear. But I can't make sense nor find a solution for (ii) and (iii).

Which is why I require your help !