International Journal of Mathematics and Mathematical Sciences
Volume 12 (1989), Issue 4, Pages 725-727
doi:10.1155/S016117128900089X
Exchange PF-rings and almost PP-rings
Department of Mathematics, University of Jordan, Amman, Jordan
Received 1 June 1988; Revised 11 August 1988
Copyright © 1989 H. Al-Ezeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let R be a commutative ring with unity. In this paper, we prove that R is an almost PP–PM–ring if and only if R is an exchange PF–ring. Let X be a completely regular Hausdorff space, and let βX be the Stone Čech compactification of X. Then we prove that the ring C(X) of all continuous real valued functions on X is an almost PP–ring if and only if X is an F–space that has an open basis of clopen sets. Finally, we deduce that the ring C(X) is an almost PP–ring if and only if C(X) is a U–ring, i.e. for each f ε C(X), there exists a unit u ε C(X) such that f=u|f|.