Localizing Systems, Module Systems, and Semistar Operations

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We present the concept of module systems for cancellative monoid. This concept is a common generalization of the notion of an ideal system (as presented by F. Halter-Koch (“Ideal Systems,” Dekker, New York, 1997)) and the notion of a semistar operation (as introduced by A. Okabe and R. Matsuda (Math. J. Toyama Univ.17 (1994), 1–21)). It allows a new insight into the connection between semistar operations and localizing systems (as developed in by M. Fontana and J. A. Huckaba (in “Commutative Rings in a Non-Noetherian Setting” (S. T. Chapman and S. Glanz, Eds.), Kluwer Academic, Dordrecht/Norwell, MA, 2000)), a general theory of flatness (including results of M. Fontana (in “Advances in Commutative Ring Theory” (D. E. Dobbs et al., Eds.), pp. 271–306, Dekker, New York, 1999) and S. Gabelli (in “Advances in Commutative Ring Theory” (D. E. Dobbs et al., Eds.), pp. 391–409, Dekker, New York, 1999)) and a new presentation of the theory of generalized integral closures.