of fractional ideals by the subgroup of principal ideals is then the ideal class group of A. In a general ring, an ideal may not be invertible (in fact, already the definition of a fractional ideal is not clear). However, over a Noetherian integral domain, it is still possible to develop some theory generalizing the situation in Dedekind domains. For example, Ch. VII of Bourbaki's Algèbre commutative gives such a theory. The ideal class group of A, when it can be defined, is closely related to the Picard group of the spectrum of A (often the two are the same; e.g., for Dedekind domains). In algebraic number theory, especially in class field theory, it is more convenient to use a generalization of an ideal class group called an idele class group. Closure operations There are several operations on ideals that play roles of closures. The most basic one is the radical of an ideal. Another is the Integral closure of an ideal. Given an irredundant primary decomposition , the intersection of 's whose radicals are minimal (don’t contain any of the radicals of other 's) is uniquely determined by ; this intersection is then called the unmixed part of . It is also a closure operation. Given ideals in a ring , the ideal is called the saturation of with respect to and is a closure operation (this notion is closely related to the study of local cohomology). See also tight closure. Reduction theory Local cohomology in ideal theory Local cohomology can sometimes be used to obtain information on an ideal. This section assumes some familiarity with sheaf theory and scheme theory. Let be a module over a ring and an ideal. Then determines the sheaf on (the restriction to Y of the sheaf associated to M). Unwinding the definition, one sees: . Here, is called the ideal transform of with respect to . See also System of parameters References Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, . Ideals (ring theory) History of mathematics Commutative algebra