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

Abstract

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 almost[Formula: 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.