Immediate Extensions of Rings and Approximation of Roots

In this work we generalize the notion of immediate extensions of valued fields introduced in Krull's work [21] to arbitrary commutative rings. We show that every ring possesses maximal immediate extensions and point out the importance of linear compactness in the investigation of maximal immediate extensions of local rings as one of the natural classes of maximally complete rings. Another important, in some sense exceptional class of maximally complete rings is that of von Neumann regular rings. A necessary and sufficient condition is given for the linear compactness of any maximal immediate extension of a valuation ring. Newton's Approximation Method is used to give a short direct proof of Hensel's Lemma and to extend a result of Lang [22] for finding simple roots of polynomials over linearly compact local rings. A somewhat weaker version of Lang's condition for the existence of simple roots whose congruence classes are multiple roots modulo the maximal ideal is given by using an idea..