Fraction-dense algebras and spaces
In the following, every ring will be commutative with identity and semiprime. An \(f\)-ring is a lattice-ordered ring in which \(a \wedge b=0\) implies \(a \wedge bc=0\) for each \(c \geq 0\). For an \(f\)-ring \(A\) let \(q(A)\) be its classical ring of quotients and \(Q(A)\) be its maximal ring of quotients. Fundamental properties of these quotient rings can be found in the book by \textit{J. Lambek}: ``Lectures on rings and modules'' (2nd edition 1976; Zbl 0365.16001). The authors discuss the problem, when \(q(A)\) is rigid in \(Q(A)\). In this case, they call the \(f\)-ring \(A\) fraction-dense. Let \(C(X)\) be the ring of all real-valued continuous functions of a Tychonoff space \(X\). If \(C(X)\) is fraction-dense, then \(X\) is called fraction-dense. By theorem 1.1 an \(f\)-ring \(A\) is fraction-dense if and only if the space \(M(A)\) of minimal prime ideals is compact and extremally disconnected. This implies that a Tychonoff space is fraction-dense if and only if its Stone-Čech compactification is fraction-dense. -- In \S2 fraction- dense spaces are studied. Among other things, the following characterization is given: A compact space \(X\) is fraction-dense if and only if every regular open subset of \(X\) densely contains a cozero-set. - - The main result of \S3 is the following: Let \(G\) be an Archimedean lattice-ordered group with order unit. If \(G\) is absolute (this means that \(\text{Min} (G)\) is compact and extremally disconnected), then its \(o\)-completion and Dedekind completion coincide.
ring of real-valued continuous functions
scientific article; zbMATH DE number 417083
scientific article