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International Journal of Mathematics and Mathematical Sciences
Volume 12, Issue 4, Pages 725-727
http://dx.doi.org/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 Hindawi Publishing Corporation. 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|.