The complete uniform ring of formal polynomials (Q2710507)

In this paper the ring \(R:=\mathbb{Z}[(X)]\) of formal polynomials is defined for a given set \(X\) of variables. The authors endow \(R\) with a uniform structure \(\mathcal{U}\) such that \((R, \mathcal{U})\) becomes a complete uniform ring. (The ring operations are uniformly continuous.) Let \(\Lambda (X)\) be the ring of symmetric functions. Then \(\Lambda(X)\) is considered as a subring of \(R\). The authors show that its uniform completion \(\widetilde{\Lambda}(X)\) is nondiscrete and metrizable. Finally it is shown that the uniform rings \(\widetilde{\Lambda}(X)\) and \(\widetilde{\Lambda}(Y)\) are isomorphic for infinite sets \(X\) and \(Y\).