Valuation Theory

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In this chapter we study valuation rings and valuations. We start with the concept of a Manis valuation of a ring K; Manis valuations behave nicely if K is a ring having large Jacobson radical. We show that a Manis valuation of K gives rise to a Manis valuation ring of K and that, conversely, every Manis valuation ring of K determines a Manis valuation of K. Also, for discrete Manis valuations we prove an approximation theorem.