On valuation rings

Abstract

In this note several definitions and results concerning valuation domains are extended to valuation rings. First we define immediate extension, maximally complete, pseudo-convergent sequence and maximal. It is demonstrated that maximal is equivalent to linearly compact and that maximal implies maximally complete. Note the three are equivalent in the case of valuation domains. We conclude by establishing the existence of a maximal completion of an arbitrary valuation ring. © 1987 Rocky Mountain Mathematics Consortium.