Maximal ideals of regulous functions are not finitely generated (Q1621594)

In real algebraic geometry, \(k\)-regulous functions on a real algebraic variety are introduced as a generalization of regular functions. It is known that the rings \(\mathcal{R}^k(\mathbb{R}^{N})\) of \(k\)-regulous functions on the affine space \(\mathbb{R}^{N}\) are not Noetherian for \(N\geq2\). The present paper considers maximal ideals in the rings \(\mathcal{R}^k(X)\) of \(k\)-regulous functions on a real algebraic variety \(X\) and proves the following theorem: Theorem. Assume that the real algebraic variety \(X\) is of dimension \(\geq2\) and \(k\in\mathbb{N}\). Then every maximal ideal of the ring \(\mathcal{R}^k(X)\) is not finitely generated.