Finite homomorphisms on rings of continuous functions.
Given the rings \(A\), \(B\) and the homomorphism \(h\) from \(A\) into \(B\), we say that \(h\) is finite in case \(B\) is a finitely generated \(A\)-module. Let \(X\), \(Y\) be compact Hausdorff spaces and \(\pi\) be a continuous mapping from \(X\) into \(Y\). The authors' main result is that the induced homomorphism \(h\) from the ring \(C(X)\) of continuous real valued functions on \(X\) into the ring \(C(Y)\) is finite if, and only if, \(\pi\) is locally injective. They extend this result to the case where \(X\), \(Y\) are realcompact and \(Y\) is normal. Then \(h\) is finite if, and only if, \(\pi\) is a closed mapping and \(X\) can be covered by a finite number of cozero sets on each of which \(\pi\) is injective. Some results of \textit{G. D. Faulkner} [Proc. Am. Math. Soc. 108, No. 2, 541--546 (1990; Zbl 0679.54023)] are obtained as particular cases of their results.
scientific article; zbMATH DE number 2056847