When is C(X) a p.p. local, a p.f. local or an r-local ring?

An element Formula: see text of a ring Formula: see text is called a p.p. (respectively, p.f.) element if the principal ideal Formula: see text is projective (respectively, flat) and a ring is called a p.p. (respectively, p.f.) ring if its every element is a p.p. (respectively, p.f.) element. As a natural generalization of these notions, a ring Formula: see text is called a p.p. (respectively, p.f.) local ring if for each Formula: see text either Formula: see text or Formula: see text is a p.p. (respectively, p.f.) element. We have shown that Formula: see text is a p.p. (respectively, p.f.) local ring if and only if at most one point of the space Formula: see text fails to be a basically disconnected point (respectively, Formula: see text-point) or equivalently, Formula: see text contains a maximal ideal Formula: see text such that each element of Formula: see text, is a p.p. (respectively, p.f.) element. A ring with a unique regular maximal ideal is called an Formula: see text-local ring. To continue this approach, we introduce a space in which at most one point fails to be an almost Formula: see text-point, namely, essential almostFormula: see text-space, and we observe that non-almost Formula: see text-spaces which are essential almost Formula: see text-spaces are exactly the spaces which guarantee that Formula: see text to be an Formula: see text-local ring. Algebraic characterizations of these spaces are given and it is shown that Formula: see text is an essential almost Formula: see text-space if and only if every regular Formula: see text-ideal of Formula: see text is prime, if and only if the smallest Formula: see text-ideal containing each regular ideal of Formula: see text is a maximal ideal. Finally, we give a diagram summarizing the relations between the spaces we studied in the paper.