(i) Prove that R=C(ℝ), the ring of all real-valued functions on ℝ under pointwise operations, is

step 1 So for this problem, we want to prove that a polynomial function is continuous at every number x

(i) Prove that R=C(ℝ), the ring of all real-valued functions...

(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.

(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.

Key Concepts

Construction of Ascending Chains of Ideals

To show a ring is non-noetherian, one often explicitly constructs an infinite sequence of ideals that never stabilizes. This technique is used in the context of rings of functions or polynomial rings with infinitely many indeterminates by tailoring ideals that 'capture' increasing complexity or support conditions.

Rings of Functions and Their Ideals

In rings formed by functions (such as continuous or smooth functions), ideals can be defined by the vanishing of functions at certain sets. The structure of these function rings frequently allows the construction of infinite chains of such ideals, which is instrumental in proving that these rings do not satisfy the Noetherian property.

Noetherian Ring

A Noetherian ring is one in which every ascending chain of ideals eventually stabilizes; that is, there exists no infinite strictly increasing sequence of ideals. This core property is essential for many results in algebra and helps prevent pathological cases in ring theory.

Ascending Chain Condition (ACC) on Ideals

The ACC is a condition placed on rings that prohibits endless growth of ideals. When a ring fails this condition, one can construct an infinite sequence of ideals where each is strictly contained in the next, demonstrating that the ring is non-noetherian.

Polynomial Rings in Infinitely Many Indeterminates

When considering polynomial rings with infinitely many variables, the inability to restrict the growth of variables directly leads to the formation of infinite ascending chains of ideals. This breakdown of finite generation, contrasted with the situation in finitely generated polynomial rings (by Hilbert’s Basis Theorem), underscores why such rings are non-noetherian.

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