Purely ring characteristic of rings of continuous functions on compact topological spaces (Q1407379)

\textit{J.-P. Delfosse} [Ann. Soc. Sci. Bruxelles, Sér. I 89, 364--368 (1975; Zbl 0304.13021)] found a ring theoretic characterization of the ring \(C(K)\) of all real-valued continuous functions on a compact space \(K\). In the paper under review; the authors replace three conditions from Delfosse's characterization by another three conditions and they show that a commutative ring \(A\) with unit \(\mathbf{1}\) is isomorphic to the ring \(C(K)\) of all real-valued continuous functions on some compact space \(K\) if and only if the following properties hold: (1) for all \(a, b\in A\), there exists \(c\in A\) such that \(a^2+ b^2= c^2\); (2) for every \(a\in A\), there exist \(b,c\in A\) such that \(a= b^2- c^2\); (3) for every \(a\in A\), there exists \((\mathbf{1}+ a^2)^{-1}\); (4) if \(a\in A\) is such that there exists a sequence \((b_n)\) in \(A\) with \(n^2(a^2+ b_n^2)= \mathbf{1}\) for each \(n\in\mathbb{N}\), then \(a= 0\); (5) for every \(a\in A\), there exist \(b\in A\) and \(n\in\mathbb{N}\) such that \(a^2+ b^2= n^2\mathbf{1}\); (6) if \((a_n)\) is a sequence in \(A\) for which there exists a sequence \((m_k)\) in \(\mathbb{N}\) such that \(k^2((a_m- a_n)^2+ b^2)= 1\) for all \(m, n\geq m_k\) and appropriate \(b= b(k, m, n)\in A\), then there exists \(a\in A\) for which there is a sequence \((n_k)\) in \(\mathbb{N}\) such that \(k^2((a-a_n)^2+c^2)= \mathbf{1}\) for all \(n\geq n_k\) and appropriate \(c= c(k, n)\in A\).