91Ó°ÊÓ

(i) Prove that \(R=C(\mathbb{R})\), the ring of all real-valued functions on \(\mathbb{R}\) under pointwise operations, is not noetherian. (ii) Recall that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a \(C^{\infty}\)-function if \(\partial^{n} f / \partial x^{n}\) exists and is continuous for all \(n\). Prove that \(R=C^{\infty}(\mathbb{R})\), the ring of all \(C^{\infty}\)-functions on \(\mathbb{R}\) under pointwise operations, is not noetherian. (iii) If \(k\) is a commutative ring, prove that \(k[X]\), the polynomial ring in infinitely many indeterminates \(X\), is not noetherian.

Let \(V\) be a vector space over a field \(k\). (i) Prove that \(V\) is a free \(k\)-module. (ii) Prove that a subset \(B\) of \(V\) is a basis of \(V\) considered as a vector space if and only if \(B\) is a basis of \(V\) considered as a free \(k\)-module.

Prove the dual of Schanuel's Lemma. Given exact sequences \(0 \rightarrow M \stackrel{i}{\rightarrow} E \stackrel{p}{\rightarrow} Q \rightarrow 0 \quad\) and \(\quad 0 \rightarrow M \stackrel{i^{\prime}}{\rightarrow} E^{\prime} \stackrel{p^{\prime}}{\rightarrow} Q^{\prime} \rightarrow 0\) where \(E\) and \(E^{\prime}\) are injective, then there is an isomorphism $$ Q \oplus E^{\prime} \cong Q^{\prime} \oplus E . $$

(i) Prove that \(\mathbb{Q}\) is a flat \(\mathbb{Z}\)-module that is not faithfully flat. (ii) Prove that an abelian group \(G\) is a faithfully flat \(\mathbb{Z}\)-module if and only if it is torsion-free and \(p G \neq G\) for all primes \(p\).