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Multiplicative ideal theory has its origin in R. Dedekind¿s multiplicative theory of algebraic numbers. The main objective of multiplicative ideal theory is to investigate the multiplicative structure of integral domains by means of ideals or certain systems of ideals. It has become a powerful tool in algebraic geometry and complex analytic geometry. The modern development of multiplicative ideal theory is based on the works of W. Krull and H. Prüfer.