Pg-Extensions and p-Extensions with applications to C(X)

Essential extensions and p-extensions have been studied for commutative rings with identity in various papers, such as [P. Bhattacharjee, M. L. Knox and W. Wm. McGovern, p-Extensions, Proceedings for the OSU-Denison Conference, AMS series Contemporary Mathematics Series (to appear); p-Embeddings, Topology Appl. 160(13) (2013) 1566- 1576; R. M. Raphael, Algebraic Extensions of Commutative Regular Rings, Canad. J. Math. 22(6) (1970) 1133-1155]. The present paper applies these concepts to certain subrings of C(X). Moreover, the paper introduces a new ring extension, called a pgextension, and determines its relation to both essential extension and p-extension. It turns out that the pg-extension R → S induces a well-defined contraction map between principal ideals P(S) and P(R).