Distributive rings of continuous functions and F-spaces
The author continues his study of rings of continuous functions with values in topological rings or fields. Let E be a separable topological associative ring with identity \(1\neq 0\) and X a topological space. We denote by \(C(X,E)\) the set of continuous functions of X into E, which forms an associative ring with respect to pointwise operations. An element f in \(C(X,E)\) is called bounded if \(f(X)\) is relatively compact in E. \(C^*(X,E)\) denotes the set of all bounded functions in \(C(X,E)\). A subset S of X is said to be \(C^*_ E\)-embedded in X if every element in \(C^*(S,E)\) is extended to an element in \(C(X,E)\). Given any element f in \(C(X,E)\), we denote by \(Z(f)\) the set of zeros of f. A subset in X is called an E-cozero set if it is the complement in X of some Z(f). X is, by definition, an EF-space if every E-cozero subset of X is \(C^*_ E\)-embedded in X. Two subsets \(A_ 1\) and \(A_ 2\) are said to be E- separated if we have \(f(A_ 1)=\{0\}\), \(f(A_ 2)=\{1\}\) for some f in \(C(X,E).\) The main result, Theorem 1, states fourteen mutually equivalent properties, a sample being given below. Theorem 1: For any continuous field (=a nondiscrete locally compact field) E and any topological space X, the following are equivalent: (1) \(C(X,E)\) is a Bezout ring, i.e. every finitely generated ideal of \(C(X,E)\) is principal; (2) the lattice of ideals of \(C(X,E)\) is distributive; (3) every ideal of \(C(X,E)\) is flat; (9) if an E-cozero subset of X is partitioned into two open sets \(A_ 1\) and \(A_ 2\), then these are E-separated; (11) X is an EF-space. Theorem 2 deals with general topological rings. For such a ring E the set \(E^ X\) of all E-valued functions on X forms a right \(C(X,E)\)-module under pointwise operations. The module \(E^ X\) is called distributive if the lattice of submodules of \(E^ X\) is distributive. It is called a Bezout module if every finitely generated submodule is cyclic. Theorem 2. (a) If every right (or left) invertible element in E is invertible, then the right \(C(X,E)\)-module \(E^ X\) is a Bezout module if and only if X is discrete and E is a right Bezout ring. (b) The \(C(X,E)\)-module \(E^ X\) is distributive if and only if X is discrete and the ring E is right distributive.
rings of continuous functions with values in topological rings or fields
scientific article; zbMATH DE number 3867999
scientific article