On characterizations of a Prüfer domain

Let D be an integral domain with quotient field K. Then it is easily seen that every invertible fractional ideal of D is finitely generated. An integral domain D is called a Prufer domain if each nonzero finitely generated ideal of D is invertible. A Prufer domain may be an example of an integral domain which would have the maximum number of characterizations in all the classes of integral domains which have been already defined in commutative algebra. The number of characterizations of a Prufer domain is already over eighty now. In this paper, we continue to study a Prufer domain and we shall give some new characterizations of a Prufer domain. In Section 1, we first collect a family of well-known characterizations of a Prufer domain which is only a part of the known characterizations of a Prufer domain and we recall some definitions and preliminary results on semistar operations and localizing systemes which will be uscd in Section 2. In Section 2, we shall give some new semistar-theoretical characterizations of a Prufer domain by the use of properties of a semistar operation and a localizing system.