Purity of the Ideal of Continuous Functions with Compact Support

&#60;P&#62;Let C(X) be the ring of all continuous real valued functions defined on a completely regular T1-space. Let CK(X) be the ideal of functions with compact support. Purity of CK(X) is studied and characterized through the subspace XL, the set of all points in X with compact neighborhoods (nbhd). It is proved that CK(X) is pure if and only if XL=&#8746;f&#8712;CK supp f. if CK(X) and CK(Y) are pure ideals, then CK(X) is isomorphic to CK(Y) if and only if XL is homeomorphic to YL. It is proved that CK(X) is pure and XL is basically disconnected if and only if for every f &#8712;CK(X), the ideal (f ) is a projective C(X)-module. Finally it is proved that if CK(X) is pure, then XL is an F'-space if and only if every principal ideal of CK(X) is a flat C(X)-module. Concrete examples exemplifying the concepts studied are given.</p

On some properties of polynomials rings

For a commutative ring with unity R, it is proved that R is a PF-ring if and only if the annihilator, annR(a), for each a ϵ R is a pure ideal in R, Also it is proved that the polynomial ring, R[X], is a PF-ring if and only if R is a PF-ring. Finally, we prove that R is a PP-ring if and only if R[X] is a PP-ring

Two properties of the power series ring

For a commutative ring with unity, A, it is proved that the power series ring A〚X〛 is a PF-ring if and only if for any two countable subsets S and T of A such that S⫅annA(T), there exists c∈annA(T) such that bc=b for all b∈S. Also it is proved that a power series ring A〚X〛 is a PP-ring if and only if A is a PP-ring in which every increasing chain of idempotents in A has a supremum which is an idempotent