On PP-Endomorphism Rings

A characterization is given of when all kernels (respectively images) of endomorphisms of a module are direct summands, a necessary condition being that the endomorphism ring itself is a left (respectively right) PP-ring. This result generalizes theorems of Small, Lenzing and Colby-Rutter and shows that R is left hereditary if and only if the endomorphism ring of every injective left module is a right PP-ring.

Exchange PF-rings and almost PP-rings

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 thatf=u|f|.

Two properties of the power series ring

For a commutative ring with unity,A, it is proved that the power series ringA〚X〛is a PF-ring if and only if for any two countable subsetsSandTofAsuch thatS⫅annA(T), there existsc∈annA(T)such thatbc=bfor allb∈S. Also it is proved that a power series ringA〚X〛is a PP-ring if and only ifAis a PP-ring in which every increasing chain of idempotents inAhas a supremum which is an idempotent.